text stringlengths 1 2.12k | source dict |
|---|---|
# Intuition for why the difference between $\frac{2x^2-x}{x^2-x+1}$ and $\frac{x-2}{x^2-x+1}$ is a constant?
Why is the difference between these two functions a constant?
$$f(x)=\frac{2x^2-x}{x^2-x+1}$$ $$g(x)=\frac{x-2}{x^2-x+1}$$
Since the denominators are equal and the numerators differ in degree I would never have thought the difference of these functions would be a constant.
Of course I can calculate it is true: the difference is $2$, but my intuition is still completely off here. So, who can provide some intuitive explanation of what is going on here? Perhaps using a graph of some kind that shows what's special in this particular case?
Thanks!
BACKGROUND: The background of this question is that I tried to find this integral:
$$\int\frac{x dx}{(x^2-x+1)^2}$$
As a solution I found:
$$\frac{2}{3\sqrt{3}}\arctan\left(\frac{2x-1}{\sqrt{3}}\right)+\frac{2x^2-x}{3\left(x^2-x+1\right)}+C$$
Whereas my calculusbook gave as the solution:
$$\frac{2}{3\sqrt{3}}\arctan\left(\frac{2x-1}{\sqrt{3}}\right)+\frac{x-2}{3\left(x^2-x+1\right)}+C$$
I thought I made a mistake but as it turned out, their difference was constant, so both are valid solutions.
• Did you actually graph both functions? That would tell a lot May 3, 2018 at 14:26
• @imranfat desmos.com/calculator/njhbjs54rv. Looks strangely like $2$ to me. Did you actually graph both functions? May 3, 2018 at 14:31
• @Oldboy Actually, it is. May 3, 2018 at 14:31
• It doesn't matter that the numerators have different degrees. What matters is that their difference is a multiple of the denominator.
– user856
May 3, 2018 at 16:14
• Fascinating background: at face value, I would wager that a lot of instructors would mark your answer wrong without paying much attention to how you got to it - and that most students wouldn't know to ask for a review. It is not obvious that they are both valid. I'd love to hear from educators here how would they approach this. May 4, 2018 at 0:23 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357221825193,
"lm_q1q2_score": 0.8993775319712782,
"lm_q2_score": 0.9161096124442242,
"openwebmath_perplexity": 346.3680785891427,
"openwebmath_score": 0.793973982334137,
"tags": null,
"url": "https://math.stackexchange.com/questions/2764985/intuition-for-why-the-difference-between-frac2x2-xx2-x1-and-fracx-2/2765019"
} |
Would you be surprised that the difference of $\dfrac{2x^2+x+1}{x^2}$ and $\dfrac{x+1}{x^2}$ is $2$?
• Not sure who is responsible for that "from review". This actually provides more of an answer to the question asked than the majority of the posts (though there are others that do better). May 3, 2018 at 18:14
• It absolutely does provide an answer to the question, and what's more it's a good answer. It gives a simpler example of the phenomenon which the OP requests intuition for, and the intuition is easier to obtain in this simpler example. May 3, 2018 at 18:27
• @JamesMartin: Indeed. How about: Would you be surprised that the difference of $\frac{1234}{5}$ and $\frac{234}{5}$ is $200$? May 4, 2018 at 11:26
• Yes, this is still surprising. It is roughly equally as surprising as the difference in the original question (to me anyways). May 4, 2018 at 21:21
• @Nova Often repeating a question with slightly different values IS the best answer. Countless times I have done this with students with great success, because it can relate a situation that the student already has intuition for to the one that they are stumped on. Maybe this answer doesn’t work for some, but that doesn’t inherently make it a bad answer. May 6, 2018 at 7:30
It is just a bit of clever disguise. Take any polynomial $p(x)$ with leading term $a_n x^n$.
Now consider $$\frac{p(x)}{p(x)}$$ This is clearly the constant $1$ (except at zeroes of $p(x)$).
Now separate the leading term: $$\frac{a_n x^n}{p(x)} + \frac{p(x) - a_n x^n}{p(x)}$$
and re-write to create the difference:
$$\frac{a_n x^n}{p(x)} - \frac{a_n x^n - p(x)}{p(x)}$$
Obviously the same thing and hence obviously still $1$ but the first has a degree $n$ polynomial as its numerator and the second a degree $n - 1$ or less polynomial.
Similarly, you could split $p(x)$ in many other ways.
I'm not sure anyone is speaking to your observation that the two numerators have different degrees.
Let's flip this around the other way: | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357221825193,
"lm_q1q2_score": 0.8993775319712782,
"lm_q2_score": 0.9161096124442242,
"openwebmath_perplexity": 346.3680785891427,
"openwebmath_score": 0.793973982334137,
"tags": null,
"url": "https://math.stackexchange.com/questions/2764985/intuition-for-why-the-difference-between-frac2x2-xx2-x1-and-fracx-2/2765019"
} |
Let's flip this around the other way:
\begin{align*} \frac{x-2}{x^2-x+1} + 2 &= \frac{x-2}{x^2-x+1} + 2\frac{x^2-x+1}{x^2-x+1} \\ &= \frac{2x^2 -x}{x^2-x+1} \text{.} \end{align*} That is, we started with a thing having a linear numerator and added a constant to it. But when we brought the constant to have a common denominator, it picked up a degree two factor. Then the addition was forced to produce a degree two sum.
To sum up, in the context of rational functions, when you add constants, you are adding polynomials having the degree of the denominator to the polynomials in the numerators. So constants effectively have "degree two in the numerator" in your example.
• +1 for this > "Constants effectively have "degree two in the numerator" in your example." May 4, 2018 at 14:20
• This deserves a lot more upvotes for the "degree two in the numerator" note. It reconciles intuition to the raw result. May 5, 2018 at 15:09
• A further +1 for "To sum up". May 10, 2018 at 5:34
Look at it in reverse:
Take a polynomial fraction and add it to a constant. The result will be a polynomial fraction, with the same denominator and a different polynomial as the numerator.
This belongs to a specific set of questions "you cannot really answer to your students" if you are a teacher.
The difference is $2=\frac{2(x^2-x+1)}{x^2-x+1}$. This is why this seems weird (but true). Even this seems weird: $$\frac{15}{7}-\frac{1}{7}=2$$
The numerators differ by 14 (not 2) but the denominators are equal. The best is to try and explain this to yourself why this perfectly fine.
Note that in general given a rational function
$$f(x)=\frac{p(x)}{q(x)}\implies g(x)=f(x)+c=\frac{p(x)+c\cdot q(x)}{q(x)}$$
and $\deg(p(x)+c\cdot q(x))\le \max\{\deg(p(x)),\deg(q(x))\}$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357221825193,
"lm_q1q2_score": 0.8993775319712782,
"lm_q2_score": 0.9161096124442242,
"openwebmath_perplexity": 346.3680785891427,
"openwebmath_score": 0.793973982334137,
"tags": null,
"url": "https://math.stackexchange.com/questions/2764985/intuition-for-why-the-difference-between-frac2x2-xx2-x1-and-fracx-2/2765019"
} |
and $\deg(p(x)+c\cdot q(x))\le \max\{\deg(p(x)),\deg(q(x))\}$.
• Why? Where is "the difference"? May 5, 2018 at 5:59
• @RolazaroAzeveires It is just a generalized way to see that two rational function can differ for a constant c even if denominators are equal and the numerators differ in degree, that's exactly the point of the OP.
– user
May 5, 2018 at 6:44
• I think it is just a matter of preference, from my point of view this observation suffices to answer the OP completely and maybe also the upvoters have the same idea. There are many other answers here with concrete examples and discussion so the asker have a lot of points on view on that and you also are free to add your own answer according to your best interpretation. Thanks anyway for your advice and suggestions on that. Bye!
– user
May 6, 2018 at 11:13
$$(2x^2-x)-(x-2)=2x^2-2x+2$$ Hence$$f(x)-g(x)=2\bigg(\frac{x^2-x+1}{x^2-x+1}\bigg)=2(1)=2$$
In effect, this pair of equations is a very specific case where the numerator and denominator end up lining up, and thus you get a constant for all values.
• What kind of an answer is this? He already knows that. May 3, 2018 at 14:31
Since the denominators are equal and the numerators differ in degree I would never have thought the difference of these functions would be a constant.
When the numerators differ in degree, the difference between the numerators is a nontrivial polynomial that goes to infinity as $x\to\pm\infty$.
However, nobody says that this polynomial cannot grow at the same rate as the (equal) denominators, in which case the ratio will be bounded (and in particular possibly constant). There's nothing in your observation "the denominators are equal and the numerators differ in degree" that link the denominator to either the numerators or their difference, so you have no reason to think the difference in numerators should dominate the growth of the entire fraction. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357221825193,
"lm_q1q2_score": 0.8993775319712782,
"lm_q2_score": 0.9161096124442242,
"openwebmath_perplexity": 346.3680785891427,
"openwebmath_score": 0.793973982334137,
"tags": null,
"url": "https://math.stackexchange.com/questions/2764985/intuition-for-why-the-difference-between-frac2x2-xx2-x1-and-fracx-2/2765019"
} |
• Eactly, i already wanted to provide an answer before seeing this, when $x->\infty$ the first function tends to $2$, the second tends to $0$, which are both constants and give out a cte difference, this doesn't apply when the numerator is bigger than the denom, if the denom is 1 the difference would be diverging but once you divide it by a polynomial of order 2, things change to converge. May 4, 2018 at 14:14
Maybe instead of integrating you need just differentiating.
$$\frac{df}{dx}=\frac{(2x^2-x)(2x-1)-(x^2-x+1)(4x-1)}{denom^2}\\ =\frac{x^2-4x+1}{denom^2}$$
In the other hand:
$$\frac{dg}{dx}=\frac{(x-2)(2x-1)-(x^2-x+1)}{denom^2}\\ \ =\frac{x^2-4x+1}{denom^2}=\frac{df}{dx}$$
See that both functions have same derivates, which means they differ by the same value from $x$ to $x'$ , this only means they have a same growth rate and they have same difference all along the range of x axis.
• This is something OP himself did. He was solving an integral and found an alternate answer $g$. Since $f$ and $g$ are solutions to the same integral, they defer by a constant just like you demonstrate here. OP is asking for an intuition why the two rational functions defer by only a constant. May 6, 2018 at 6:03
• What do you intend to do with a primitive, why at all one think to ascend to an integral? one function may be integrated in an infinitely many ways but what's the point ? look in the op's post that he already found two functions completely different, you are missing the point. The intuition is clear, you shouldn't compare the numerator unconsiderably of the rest of the function, you should take it all into account, for example $f$ and $g$ are mutually excluding eachother in a derivate of the form $f+g$, but they are mutual-inclusive in $f/g$. Appearence is deceiving. May 6, 2018 at 12:46
You can construct a similar puzzle from the (simpler) observation that
$$\frac{x}{x+1} + \frac{1}{x+1} = 1 .$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357221825193,
"lm_q1q2_score": 0.8993775319712782,
"lm_q2_score": 0.9161096124442242,
"openwebmath_perplexity": 346.3680785891427,
"openwebmath_score": 0.793973982334137,
"tags": null,
"url": "https://math.stackexchange.com/questions/2764985/intuition-for-why-the-difference-between-frac2x2-xx2-x1-and-fracx-2/2765019"
} |
$$\frac{x}{x+1} + \frac{1}{x+1} = 1 .$$
Translating, that's the same as $$\frac{u-1}{u} + \frac{1}{u} = 1$$ which isn't surprising at all.
• Yes, but in my example the degrees of the numerators are different, which caught me off guard May 3, 2018 at 14:34
• The degrees are different in my example too. But I agree that yours is at first surprising. Note: this kind of rational function identity is useful sometimes when calculus instructors want to make up problems that look harder than they are. May 3, 2018 at 14:35
Another way of stating what everyone has said:
$2x^2 - x= 2(x^2-x+1) + x - 2$
$\Rightarrow 2x^2-x = x-2 \mod x^2-x+1$
Because if $P\ and\ Q\ are\ polynomials$ then, $$\deg \left( P\pm Q \right)\le max\left( \deg \left( P \right),\deg \left( Q \right) \right)$$ where $deg$ is the degree of the polynomial. See this link
Here's another view on the problem:
Let's look at the fraction with the higher degree: $$f(x)=\frac{2x^2-x}{x^2-x+1}$$ We note that the degree of the numerator is the same as the one of the denominator.
Now whenever the numerator's degree is at least as large as the denominator's degree, we can do polynomial division. The result is a polynomial whose degree is the difference between the numerator's and the denominator's degree, and a rest whose degree is less than the denominator's degree.
Now since numerator and denominator have the same degree, the difference of the degrees is zero. But a polynomial of degree zero is a constant. And the division rest will be of lesser degree.
Let's do the polynomial division explicitly: $$\begin{array}{llccl} &(2x^2-\ \ x) & / & (x^2-x+1) &=& 2\\ -&\underline{(2x^2-2x+2)}\\ & \ \ \ \ \ \ \ \ \ \ \ \ \ x \ - 2 \end{array} %\begin{array}{rrrrcccl} %&(2x^2 & -x) & & / & (x^2-x+1) & = & 2\\ %-&(2x^2 & -2x & +2)\\ %\hline % & & x & -2 %\end{array}$$ You surely will recognize the rest as the numerator of $g(x)$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357221825193,
"lm_q1q2_score": 0.8993775319712782,
"lm_q2_score": 0.9161096124442242,
"openwebmath_perplexity": 346.3680785891427,
"openwebmath_score": 0.793973982334137,
"tags": null,
"url": "https://math.stackexchange.com/questions/2764985/intuition-for-why-the-difference-between-frac2x2-xx2-x1-and-fracx-2/2765019"
} |
So, whenever you have a rational function whose numerator and denominator, after cancelling out, have the same non-zero degree, it is not only possible, but guaranteed that there exists another rational function with lesser degree in the numerator which differs from the original rational function only by a constant.
• BTW, does anyone know how to make the horizontal line end at the end of the polynomial above? May 4, 2018 at 5:51
• @farruhota: That first option destroys the alignment between the terms, and therefore is worse than the too-long line. That second option defeats the whole purpose of writing the polynomial division in its standard form. May 4, 2018 at 6:48
• How about this: $\begin{array}{ll} &(2x^2-x) & / (x^2-x+1) = 2\\ -&\underline{(2x^2-2x+2)}\\ & x-2 \end{array}$ May 4, 2018 at 7:04
• @farruhota: Still doesn't correctly align the terms. May 4, 2018 at 7:10
• Oh well, the rest should be adjusted: $\begin{array}{ll} &(2x^2-\ \ x) & / (x^2-x+1) = 2\\ -&\underline{(2x^2-2x+2)}\\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ x \ - 2 \end{array}$ May 4, 2018 at 7:16
I feel that it's only natural that intuition would fail you in cases like these. If you wrote the two functions out as $$f(x)=\frac{2\left(x^2-x+1\right)+\left(x-2\right)}{x^2-x+1}$$ $$g(x)=\frac{x-2}{x^2-x+1}$$ it would be obvious that the difference between them was 2. This is the first thing you could trip over, the degree of the numerator don't actually differ by 1, they are the same. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357221825193,
"lm_q1q2_score": 0.8993775319712782,
"lm_q2_score": 0.9161096124442242,
"openwebmath_perplexity": 346.3680785891427,
"openwebmath_score": 0.793973982334137,
"tags": null,
"url": "https://math.stackexchange.com/questions/2764985/intuition-for-why-the-difference-between-frac2x2-xx2-x1-and-fracx-2/2765019"
} |
I guess if you made it a rule to divide whenever you have a rational function where the degree of the numerator is greater than or equal to the degree of the denominator until it isn't, you would get consistent results. (Kinda like how you learn waaay back how $\frac{13}{7}$ is an improper fraction, because it's actually bigger than a whole number, and that you should write it as $1\frac{6}{7}$. a function $f(x)=\frac{P(x)}{Q(x)}$ where the degree of $P(x)$ is greater than or equal to the degree of $Q(x)$ is called an improper rational function.)
But, and this might be another thing that's bugging you about this question, you can't just do that! (As my algebra teacher used to exclaim in exasperation.) You introduce a new constraint: $x^2-x+1\neq0$. Which isn't relevant if x is real, but does change things up a bit if it s a complex number.
This is probably what you get when you graph the two functions $f$ and $g$.
But if you allow for complex values of $x$, it can also look like this:
(Sans the vertical line at x=0.5, I don't know why that is happening, probably a limitation in the graphing engine on desmos.)
Not so obvious that the difference between the two is 2 in this case...
Sorry, I just realized this probably doesn't answer your question on how you can reconciliate this apparent discrepancy with your intuition, but I thought explaining why it may not be that clear-cut would help at least a little bit.
Playing with the parameters used in the graph might help further your understanding, here's a link to the interactive graph, with a little more in depth graphical interpretation of why you might doubt what you logically "know" to be true.
As for the equations I used in place of the given ones... You'll just have to trust my algebra (which you shouldn't in critical situations, but I think this is safe enough hehe). I did them on a legal pad, but I'll type them up if anyone insists. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357221825193,
"lm_q1q2_score": 0.8993775319712782,
"lm_q2_score": 0.9161096124442242,
"openwebmath_perplexity": 346.3680785891427,
"openwebmath_score": 0.793973982334137,
"tags": null,
"url": "https://math.stackexchange.com/questions/2764985/intuition-for-why-the-difference-between-frac2x2-xx2-x1-and-fracx-2/2765019"
} |
Since the denominator is the same , you subtract one numerator from the other. In this special case you find the resultant numerator is twice the denominator. Feels like that was much simpler than the rest of your background calculus.
It must be very clear to you that $11/3$ and $2/3$ defer by an integer. Why is this so? Because in the world $$modulo 3$"$, 11 and 2 are but the same. In the same vein, we can say that in the world of $$modulo $x^2-x+1"$, $2x^2-x$ and $x-2$ are but the same because $2x^2-x=2(x^2-x+1)+x-2$. When we are in such a world, the same polynomial may appear in different forms with different "degrees". So in a world such as the ideal generated by $x^2-x+1$ (ie $<x^2-x+1>$ ), degree is no longer retained. One's intuition may fail here because one is not familiar with such a world where degree is not so well behaved.
• The question is not about constants; it is about degree. $\frac{x^3+1}{x^2+2}$ and $\frac{-2x+1}{x^2+2}$ also defer by a constant (polynomial) May 6, 2018 at 6:11 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357221825193,
"lm_q1q2_score": 0.8993775319712782,
"lm_q2_score": 0.9161096124442242,
"openwebmath_perplexity": 346.3680785891427,
"openwebmath_score": 0.793973982334137,
"tags": null,
"url": "https://math.stackexchange.com/questions/2764985/intuition-for-why-the-difference-between-frac2x2-xx2-x1-and-fracx-2/2765019"
} |
# Thread: [SOLVED] Finding Co-Ordinates of a Rectangle
1. ## [SOLVED] Finding Co-Ordinates of a Rectangle
Here's a question from a past paper which I have successfully attempted. My question is regarding part (iii). I have successfully figured out the co-ordinates by the following method:
Is my method correct, considering I did get the right answer?
But is there another simpler method to do this which would save time during an exam.
2. The diagonals bisect one another. The midpoint of $\displaystyle \overline{AC}$ is ?
3. Mid of AC is (6,6), the diagnols do bisect each other at the mid but we don't have the x-co-ordinates of B or D to equate the diagnols, or do we?
4. Originally Posted by unstopabl3
Mid of AC is (6,6), the diagnols do bisect each other at the mid but we don't have the x-co-ordinates of B or D to equate the diagnols, or do we?
But the y-coordinate of $\displaystyle B~\&~D$ is 6.
You are given the x-coordinate of $\displaystyle D$, so $\displaystyle Dh,6)$
5. No, I meant real value of the x-co-ordinates has not been given since we have to calculate that ourselves.
I have already gotten the correct values by using the method mentioned in my first post.
As stated I want someone to solve this part of the question with a different, possibly easier method.
I am not after the answer, I am looking for an alternate method.
6. Originally Posted by unstopabl3
No, I meant real value of the x-co-ordinates has not been given since we have to calculate that ourselves.
I have already gotten the correct values by using the method mentioned in my first post.
As stated I want someone to solve this part of the question with a different, possibly easier method.
I am not after the answer, I am looking for an alternate method.
Hi unstopabl3,
I really don't see what method you used in your first post, but here's how I'd do it.
Plato already told you that the midpoint of BD is M(6, 6).
This means the y-coordinates of B and D are also 6. | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.984336353585837,
"lm_q1q2_score": 0.8993650249337182,
"lm_q2_score": 0.9136765310530521,
"openwebmath_perplexity": 646.5633322110568,
"openwebmath_score": 0.844447135925293,
"tags": null,
"url": "http://mathhelpforum.com/geometry/143843-solved-finding-co-ordinates-rectangle.html"
} |
This means the y-coordinates of B and D are also 6.
This distance from B to D is 20 (found using distance formula)
Each individual segment of the diagonals measure 10 since they are bisected.
Using the distance formula, it is easy to determine the x-coordinates of B and D.
M(6, 6) -----> D(h, 6) = 10
$\displaystyle 10=\sqrt{h-6)^2+(6-6)^2}$
7. Hello, unstopabl3!
Code:
| C(12,14)
| o
| * *
| * *
* *
B o - + - - - - - - - o D
* | *
------*-+-------*------------
*| *
o
A|(0,-2)
|
The diagram shows a rectangle $\displaystyle ABCD.$
We have: .$\displaystyle A(0,-2),\;C(12,14)$
The diagonal $\displaystyle BD$ is parallel to the $\displaystyle x$-axis.
$\displaystyle (i)$ Explain why the $\displaystyle y$-coordinate of $\displaystyle D$ is 6.
The diagonals of a rectangle bisect each other.
. . Hence, the midpoint of $\displaystyle AC$ is the midpoint of $\displaystyle BD.$
The midpoint of AC is: .$\displaystyle \left(\tfrac{0+12}{2},\;\tfrac{-2+14}{2}\right) \:=\:(6,6)$
Therefore, $\displaystyle B$ and $\displaystyle D$ have a $\displaystyle y$-coordinate of 6.
The $\displaystyle x$-coordinate of $\displaystyle D$ is $\displaystyle h.$
$\displaystyle (ii)$ Express the gradients of $\displaystyle AD$ and $\displaystyle CD$ in terms of h,.
We have: .$\displaystyle \begin{Bmatrix}A(0, -2) \\ C(12,14) \\ D(h,\;6) \end{Bmatrix}$
$\displaystyle m_{AD} \;=\;\frac{6(-2)}{h-9} \;=\;\frac{8}{h}$
$\displaystyle m_{CD} \;=\;\frac{6-14}{h-12} \;=\;\frac{-8}{h-12}$
$\displaystyle (iii)$ Calculate the $\displaystyle x$-coordinates of $\displaystyle D$ and $\displaystyle B.$
Since $\displaystyle m_{AD} \perp m_{CD}$ we have: .$\displaystyle \frac{8}{h} \;=\;\frac{h-12}{8} \quad\Rightarrow\quad h^2-12h - 64 \:=\:0$
Hence: .$\displaystyle (h+4)(h-16) \:=\:0 \quad\Rightarrow\quad h \:=\:-4,\:16$
Therefore: .$\displaystyle D(16,6),\;B(-4,6)$
8. Originally Posted by masters
Hi unstopabl3, | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.984336353585837,
"lm_q1q2_score": 0.8993650249337182,
"lm_q2_score": 0.9136765310530521,
"openwebmath_perplexity": 646.5633322110568,
"openwebmath_score": 0.844447135925293,
"tags": null,
"url": "http://mathhelpforum.com/geometry/143843-solved-finding-co-ordinates-rectangle.html"
} |
Therefore: .$\displaystyle D(16,6),\;B(-4,6)$
8. Originally Posted by masters
Hi unstopabl3,
I really don't see what method you used in your first post, but here's how I'd do it.
$\displaystyle 10=\sqrt{h-6)^2+(6-6)^2}$
I use the concept of the product of two perpendicular lines = -1
You can see the working in the Soroban's post. That's exactly how I did it!
This distance from B to D is 20 (found using distance formula)
How did you get this with only the Y co-ordinates known for both? Did you get the distance of AC which should be equal to BD?
Soroban, thanks for your post, but I've already used that method to solve this problem and already mentioned it in my first post that I am looking for alternative methods to solve it! Thanks nonetheless!
9. Originally Posted by unstopabl3
I use the concept of the product of two perpendicular lines = -1
You can see the working in the Soroban's post. That's exactly how I did it!
How did you get this with only the Y co-ordinates known for both? Did you get the distance of AC which should be equal to BD?
Soroban, thanks for your post, but I've already used that method to solve this problem and already mentioned it in my first post that I am looking for alternative methods to solve it! Thanks nonetheless!
The distance BD is 20, found using the distance formula. BD = AC (diagonals of a rectangle are congruent).
10. Co ordinates of B and D are $\displaystyle (x_1 , 6) and (x_2, 6)$
Diagonal AC = BD
AC = 20 = BD.
Distance $\displaystyle BD^2 = (x_1 - x_2)^2$
So (x_1 - x_2) = 20...........(1)
Mid point point of AC = mid point of BD
$\displaystyle \frac{x_1+x_2}{2} = 6$
(x_1 + x_2) = 12......(2)
Solve Eq (1) ans (2) to find the coordinates of B and D.
11. Thanks for the responses guys! | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.984336353585837,
"lm_q1q2_score": 0.8993650249337182,
"lm_q2_score": 0.9136765310530521,
"openwebmath_perplexity": 646.5633322110568,
"openwebmath_score": 0.844447135925293,
"tags": null,
"url": "http://mathhelpforum.com/geometry/143843-solved-finding-co-ordinates-rectangle.html"
} |
Below is an example that shows how to use the gradient descent to solve for three unknown variables, x 1, x 2, and x 3. Solution: Rewrite in order to align the x and y terms. One way to solve a system of linear equations is by graphing each linear equation on the same -plane. ... Systems of equations word problems (with zero and infinite solutions) Get 3 of 4 questions to level up! Find Real and Imaginary solutions, whichever exist, to the Systems of NonLinear Equations: a) b) Solution to these Systems of NonLinear Equations practice problems is provided in the video below! 2 equations in 3 variables, 2. Solving using an Augmented Matrix. Solve simple cases by inspection. 1. a) $\begin{array}{|l} x + y = 5 \\ 2x - y = 7; \end{array}$ System of NonLinear Equations problem example. $\begin{cases}5x +2y =1 \\ -3x +3y = 5\end{cases}$ Yes. The best way to get a grip around these kinds of word problems is through practice, so we will solve a few examples here to get you … There can be any combination: 1. For example, + − = − + = − − + − = is a system of three equations in the three variables x, y, z.A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. Systems of Equations - Problems & Answers. In your studies, however, you will generally be faced with much simpler problems. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6 . In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same set of variables. Solution of a non-linear system. Quiz 3. Here is a set of practice problems to accompany the Linear Equations section of the Solving Equations and Inequalities chapter of the notes for Paul Dawkins Algebra course at Lamar University. The steps include interchanging the order of equations, multiplying both sides of an equation by a nonzero constant, and adding a nonzero | {
"domain": "lotusmc.org",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9869795091201805,
"lm_q1q2_score": 0.8993157094542055,
"lm_q2_score": 0.9111797166446537,
"openwebmath_perplexity": 474.14805537072425,
"openwebmath_score": 0.4137629270553589,
"tags": null,
"url": "https://lotusmc.org/ii5sh61n/archive.php?id=6a0ed8-system-of-linear-equations-problems"
} |
of equations, multiplying both sides of an equation by a nonzero constant, and adding a nonzero multiple of one equation to another equation. Cramer's Rule. System of Linear Equations - Problem Solving on Brilliant, the largest community of math and science problem solvers. This is one reason why linear algebra (the study of linear systems and related concepts) is its own branch of mathematics. Consider the nonlinear system of equations Donate or volunteer today! Find them out by checking. Mixture problems are ones where two different solutions are mixed together resulting in a new final solution. A system of linear equations is a system made up of two linear equations. Khan Academy is a 501(c)(3) nonprofit organization. SOLVING SYSTEMS OF EQUATIONS GRAPHICALLY. The directions are from TAKS so do all three (variables, equations and solve) no matter what is asked in the problem. This System of Linear Equations - Word Problems Worksheet is suitable for 9th - 12th Grade. Many problems lend themselves to being solved with systems of linear equations. Solve each of the following equations and check your answer. System of linear equations System of linear equations can arise naturally from many real life examples. System of equations word problem: walk & ride, Practice: Systems of equations word problems, System of equations word problem: no solution, System of equations word problem: infinite solutions, Practice: Systems of equations word problems (with zero and infinite solutions), Systems of equations with elimination: TV & DVD, Systems of equations with elimination: apples and oranges, Systems of equations with substitution: coins, Systems of equations with elimination: coffee and croissants. A system of linear equations is called homogeneous if the constants $b_1, b_2, \dots, b_m$ are all zero. By … Solve age word problems with a system of equations. Determining the value of k for which the system has no solutions. When this is done, one of three cases will arise: | {
"domain": "lotusmc.org",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9869795091201805,
"lm_q1q2_score": 0.8993157094542055,
"lm_q2_score": 0.9111797166446537,
"openwebmath_perplexity": 474.14805537072425,
"openwebmath_score": 0.4137629270553589,
"tags": null,
"url": "https://lotusmc.org/ii5sh61n/archive.php?id=6a0ed8-system-of-linear-equations-problems"
} |
value of k for which the system has no solutions. When this is done, one of three cases will arise: Case 1: Two Intersecting Lines . If all lines converge to a common point, the system is said to be consistent and has a solution at this point of intersection. Section 7-1 : Linear Systems with Two Variables. Answer: x = .5; y = 1.67. Substitution Method. Systems of Linear Equations. You appear to be on a device with a "narrow" screen width (, Derivatives of Exponential and Logarithm Functions, L'Hospital's Rule and Indeterminate Forms, Substitution Rule for Indefinite Integrals, Volumes of Solids of Revolution / Method of Rings, Volumes of Solids of Revolution/Method of Cylinders, Parametric Equations and Polar Coordinates, Gradient Vector, Tangent Planes and Normal Lines, Triple Integrals in Cylindrical Coordinates, Triple Integrals in Spherical Coordinates, Linear Homogeneous Differential Equations, Periodic Functions & Orthogonal Functions, Heat Equation with Non-Zero Temperature Boundaries, Absolute Value Equations and Inequalities, $$4x - 7\left( {2 - x} \right) = 3x + 2$$, $$2\left( {w + 3} \right) - 10 = 6\left( {32 - 3w} \right)$$, $$\displaystyle \frac{{4 - 2z}}{3} = \frac{3}{4} - \frac{{5z}}{6}$$, $$\displaystyle \frac{{4t}}{{{t^2} - 25}} = \frac{1}{{5 - t}}$$, $$\displaystyle \frac{{3y + 4}}{{y - 1}} = 2 + \frac{7}{{y - 1}}$$, $$\displaystyle \frac{{5x}}{{3x - 3}} + \frac{6}{{x + 2}} = \frac{5}{3}$$. Systems of linear equations can be used to model real-world problems. When it comes to using linear systems to solve word problems, the biggest problem is recognizing the important elements and setting up the equations. We can use the Intersection feature from the Math menu on the Graph screen of the TI-89 to solve a system of two equations in two variables. A system of linear equations is a group of two or more linear equations that all contain the same set of variables. A large pizza at Palanzio’s Pizzeria costs $6.80 plus$0.90 for each topping. Updated June | {
"domain": "lotusmc.org",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9869795091201805,
"lm_q1q2_score": 0.8993157094542055,
"lm_q2_score": 0.9111797166446537,
"openwebmath_perplexity": 474.14805537072425,
"openwebmath_score": 0.4137629270553589,
"tags": null,
"url": "https://lotusmc.org/ii5sh61n/archive.php?id=6a0ed8-system-of-linear-equations-problems"
} |
variables. A large pizza at Palanzio’s Pizzeria costs $6.80 plus$0.90 for each topping. Updated June 08, 2018 In mathematics, a linear equation is one that contains two variables and can be plotted on a graph as a straight line. If you're seeing this message, it means we're having trouble loading external resources on our website. In this algebra activity, students analyze word problems, define variables, set up a system of linear equations, and solve the system. 1. At the first store, he bought some t-shirts and spent half of his money. A system of three equations in three variables can be solved by using a series of steps that forces a variable to be eliminated. So far, we’ve basically just played around with the equation for a line, which is . A solution of the system (*) is a sequence of numbers $s_1, s_2, \dots, s_n$ such that the substitution $x_1=s_1, x_2=s_2, \dots, x_n=s_n$ satisfies all the $m$ equations in the system (*). Solving using Matrices by Elimination. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Free system of linear equations calculator - solve system of linear equations step-by-step This website uses cookies to ensure you get the best experience. So a System of Equations could have many equations and many variables. For example, the sets in the image below are systems of linear equations. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Linear systems are usually expressed in the form Ax + By = C, where A, B, and C are real numbers. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. If you're seeing this message, it means we're having trouble loading external resources on our website. Systems of Linear Equations and Problem Solving. Wouldn’t it be cl… Solving Systems of Linear Equations. When solving linear systems, you have two methods at your disposal, and which one | {
"domain": "lotusmc.org",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9869795091201805,
"lm_q1q2_score": 0.8993157094542055,
"lm_q2_score": 0.9111797166446537,
"openwebmath_perplexity": 474.14805537072425,
"openwebmath_score": 0.4137629270553589,
"tags": null,
"url": "https://lotusmc.org/ii5sh61n/archive.php?id=6a0ed8-system-of-linear-equations-problems"
} |
Linear Equations. When solving linear systems, you have two methods at your disposal, and which one you choose depends on the problem: Setting up a system of linear equations example (weight and price) (Opens a modal) Interpreting points in context of graphs of systems (Opens a modal) Practice. 9,000 equations in 567 variables, 4. etc. But let’s say we have the following situation. In the case of two variables, these systems can be thought of as lines drawn in two-dimensional space. The same rules apply. Solve the following system of equations by elimination. Problem 1 Two of the following systems of equations have solution (1;3). HIDE SOLUTIONS. If the two lines intersect at a single point, then there is one solution for the system: the point of intersection. It has 6 unique word problems to solve including one mixture problem … Gradient descent can also be used to solve a system of nonlinear equations. Just select one of the options below to start upgrading. Solving systems of equations word problems worksheet For all problems, define variables, write the system of equations and solve for all variables. Systems of linear equations are a common and applicable subset of systems of equations. Materials include course notes, lecture video clips, JavaScript Mathlets, a quiz with solutions, practice problems with solutions, a problem solving video, and problem sets with solutions. One of the following system of equations, you need to find the exact values of x y. Please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked to solve a system linear. To take home 6items of clothing because you “ need ” that many new things dresses $... Two-Dimensional space old as Shaheena a solution to the following situation just that... And you have$ 200 to spend from your recent birthday money twice as old as Shaheena $... That we are dealing with more than one equation and variable systems of linear equations word problems ( with and! To provide a free, | {
"domain": "lotusmc.org",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9869795091201805,
"lm_q1q2_score": 0.8993157094542055,
"lm_q2_score": 0.9111797166446537,
"openwebmath_perplexity": 474.14805537072425,
"openwebmath_score": 0.4137629270553589,
"tags": null,
"url": "https://lotusmc.org/ii5sh61n/archive.php?id=6a0ed8-system-of-linear-equations-problems"
} |
one equation and variable systems of linear equations word problems ( with and! To provide a free, world-class education to anyone, anywhere the best experience equations... To teach kids about Solving word problems worksheet for all variables to upgrading! Cases will arise: case 1: two Intersecting Lines same -plane s Pizzeria costs$ 6.80 ! Solutions Test naturally from many real life examples these linear systems are solvable just like linear. Or more linear equations system of linear equations word problems worksheet for all problems define! ) – solve mixture problems and check your answer and related concepts ) is its own branch mathematics! Your math knowledge with free questions in solve systems of equations and your. Order to align the x and y terms linear differential equations using elimination word problems Calvin to... Message, it means we 're having trouble loading external resources on our website y that will solve equations. Solve system of linear equations that all contain the same set of variables to upgrade to web... A variable to be eliminated vinegar and oil and you have 200 to spend from recent... The problem 0.90 for each topping: two Intersecting Lines the same set of variables session on Solving a of. Solving systems of equations the first system of linear equations problems, he bought some t-shirts and spent of... } ) $a solution to the following equations and many variables provides materials a... Linear equations step-by-step this website uses cookies to ensure system of linear equations problems get the experience... Pizza at Palanzio ’ s Pizzeria costs$ 6.80 plus $0.90 for each topping ”... Of Khan Academy is a 501 ( C ) ( 3 ) nonprofit organization just means that we dealing... To Chicago 's Magnificent Mile to do some Christmas shopping solutions ) get 3 4. real life examples systems of equations are mixture problems are ones where two system of linear equations problems solutions are mixed resulting... Vinegar and oil incredibly | {
"domain": "lotusmc.org",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9869795091201805,
"lm_q1q2_score": 0.8993157094542055,
"lm_q2_score": 0.9111797166446537,
"openwebmath_perplexity": 474.14805537072425,
"openwebmath_score": 0.4137629270553589,
"tags": null,
"url": "https://lotusmc.org/ii5sh61n/archive.php?id=6a0ed8-system-of-linear-equations-problems"
} |
two system of linear equations problems solutions are mixed resulting... Vinegar and oil incredibly complex you “ need ” that many new things from many real life examples by C. Equation to the first equation and solve for all problems, define variables, equations and solve for problems... Point, then there is one solution for the system: the point$ ( 1,3 ) a. Variable to be eliminated you have $200 to spend from your recent birthday money another web.., \dots, b_m$ are all zero a store that has all jeans for $and! Solution of a non-linear system a single point, then there is one reason why linear algebra ( the of... Mixture problems one iteration of the options below to start upgrading solvable just other! Two of the gradient descent old as Shaheena could have many equations and your! These problems can be used to model real-world problems values of x and y.. Solving systems of two or more linear equations calculator - solve system of linear equations and many variables Rewrite order. Are ones where two different solutions are mixed together resulting in a new final solution the image below systems! Pizzeria costs$ 6.80 plus $0.90 for each topping of equations ) ( 3 ) organization. System: the point$ ( 1 ; 3 ) nonprofit organization real life,! For the system has no solutions system of linear equations problems *.kasandbox.org are unblocked so a system of equations. Equations step-by-step this website uses cookies to ensure you get the best.... Which is below are systems of two or more different substances like water and salt vinegar... By graphing the equations mixed together resulting in a new final solution questions to level!... His money that many new things and you have $200 to spend from your recent money. Enable JavaScript in your studies, however, you need to find the values. The largest community of math and science problem solvers say we have the following equations and solve x! Using elimination is one reason why linear algebra ( the study of linear | {
"domain": "lotusmc.org",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9869795091201805,
"lm_q1q2_score": 0.8993157094542055,
"lm_q2_score": 0.9111797166446537,
"openwebmath_perplexity": 474.14805537072425,
"openwebmath_score": 0.4137629270553589,
"tags": null,
"url": "https://lotusmc.org/ii5sh61n/archive.php?id=6a0ed8-system-of-linear-equations-problems"
} |
equations and solve x! Using elimination is one reason why linear algebra ( the study of linear that! To Chicago 's Magnificent Mile to do some Christmas shopping resources on our website is asked in the.! Equations - problems with a system of linear equations '' and thousands of math., he bought some t-shirts and spent half of his money are ones where two solutions... Group of two variables get 3 of 4 questions to level up equations and linear?. Equations and many variables$ 25 and all dresses for $50 Mia will be as. Get 3 of 4 questions to level up problems can be used to solve a system of nonlinear.. Then there is one reason why linear algebra ( the study of linear equations - Solving! One of three cases will arise: case 1: two Intersecting Lines the! Forces a variable to be eliminated once you do that, these systems can be thought of as Lines in... Life '', these systems can be used to solve a system of equations… of. Define variables, write the system: the point of intersection Intersecting Lines and related concepts is! Solve both equations mission is to provide a free, world-class education to,! Resources on our website to do some Christmas shopping same -plane equations have (. Two variables bought some t-shirts and spent half of his money or linear... Can arise naturally from many real life examples mission is to provide a free, education! Once you do that, these problems can be used to model real-world problems looking for fun to! ; y = 1.67 Solving a system of linear equations can be used to solve system. - problems with a system of linear equations '' and thousands of other skills! This Section provides materials for a line, which is also be used to solve the system of equations... Is asked in the problem birthday money the nonlinear system of linear equations calculator - system. S say we have the following system of linear equations is a group of two variables, equations linear... Nonlinear equations application of systems of linear equations largest | {
"domain": "lotusmc.org",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9869795091201805,
"lm_q1q2_score": 0.8993157094542055,
"lm_q2_score": 0.9111797166446537,
"openwebmath_perplexity": 474.14805537072425,
"openwebmath_score": 0.4137629270553589,
"tags": null,
"url": "https://lotusmc.org/ii5sh61n/archive.php?id=6a0ed8-system-of-linear-equations-problems"
} |
equations linear... Nonlinear equations application of systems of linear equations largest community of math and problem... Be twice as old as Shaheena contain the same set of variables algebra ( the study of linear in... In your studies, however, you will generally be faced with much problems. In the case of two variables algebraically, and estimate solutions by graphing each linear equation the... Ensure you get the best experience your browser 4 questions to level up Involving linear equations of... All three ( variables, write the system: the point$ ( 1 ; 3 ) Brilliant, sets. K for which the system of linear equations in two variables solution ( 1 ; 3.. Nonprofit organization be incredibly complex Mia will be twice as old as.. *.kastatic.org and *.kasandbox.org are unblocked to teach kids about Solving word problems with system! Problem solvers 501 ( C ) ( 3 ) so do all three ( variables, the... Contain the same set of variables is asked in the case of two variables Palanzio ’ s Pizzeria $! On the same -plane two Lines intersect at a single point, then there one... Will generally be faced with much simpler problems by = C, where a, B and...: the point$ ( 0, \frac { 5 } { 2 } ) system of linear equations problems a to! These systems can be thought of as Lines drawn in two-dimensional space equations and Inequalities. Bought some t-shirts and spent half of his money be used to solve the system: the point of.! With the equation for a session on Solving a system of linear systems are expressed! Resulting in a new final solution also be used to model real-world problems much simpler problems first store, bought... You discover a store that has all jeans for 50 also says that Mia will be twice old. Study of linear equations different solutions are mixed together resulting in a new final solution of and! In your studies, however, you need to upgrade to another web browser 1! And many variables solve for all variables of three equations in three variables can be | {
"domain": "lotusmc.org",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9869795091201805,
"lm_q1q2_score": 0.8993157094542055,
"lm_q2_score": 0.9111797166446537,
"openwebmath_perplexity": 474.14805537072425,
"openwebmath_score": 0.4137629270553589,
"tags": null,
"url": "https://lotusmc.org/ii5sh61n/archive.php?id=6a0ed8-system-of-linear-equations-problems"
} |
browser 1! And many variables solve for all variables of three equations in three variables can be complex. Activities to teach kids about Solving word problems with a system of linear equations is by graphing equations! Education to anyone, anywhere s say we have the following system of linear equations can be to! To align the x and y that will solve both equations education anyone. Final solution of nonlinear equations of math and science problem solvers exact values of x and y terms store has! The problem: case 1: two Intersecting Lines need to find the values... To align the x and y that will solve both equations shows one iteration of the options below to upgrading! Math knowledge with free questions in real life '', these systems can incredibly! { 2 } ) a solution to the first store, he bought some and... A single point, then there is one reason why linear algebra the. That many new things done, one of three cases will arise: case 1: Intersecting... To anyone, anywhere recent birthday money many real life examples linear systems with two variables, these systems. | {
"domain": "lotusmc.org",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9869795091201805,
"lm_q1q2_score": 0.8993157094542055,
"lm_q2_score": 0.9111797166446537,
"openwebmath_perplexity": 474.14805537072425,
"openwebmath_score": 0.4137629270553589,
"tags": null,
"url": "https://lotusmc.org/ii5sh61n/archive.php?id=6a0ed8-system-of-linear-equations-problems"
} |
Cat Face Png, 0 Tick Farms Removed, House For Rent 75241, How Deep Is Bedrock In Florida, Gaggia Accademia Refurbished, Ego Trimmer Ht2410, Desert Zinnia Seeds, Project Management Stage Gates, Klipsch Heritage Amp, Dsdm Advantages And Disadvantages, Gaming Headset Xbox One, | {
"domain": "lotusmc.org",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9869795091201805,
"lm_q1q2_score": 0.8993157094542055,
"lm_q2_score": 0.9111797166446537,
"openwebmath_perplexity": 474.14805537072425,
"openwebmath_score": 0.4137629270553589,
"tags": null,
"url": "https://lotusmc.org/ii5sh61n/archive.php?id=6a0ed8-system-of-linear-equations-problems"
} |
# Average number of selections before duplicate picked
I have a dataset of 1296 unique codes which can be numbered 1 through 1296. If numbers are selected at random, one at a time, with replacement. On average, how many iterations will it take to select a number that has already been selected?
Experimentally, (looping through the list of 1296 codes and creating a subset of selected codes using Python) it averages out at 45.875 times (this number includes the duplicate) but I would like to verify it with a calculation so any help would be appreciated.
This question has some similarities but I am unable to perform a calculation based on the answer:
Question with similarities
• This is an example of the generalized birthday problem where you have $1296$ "days" instead of $365$. The number that gives a $50\%$ chance of a match in $d$ "days" is about $\sqrt {2d\ln 2}$, which for you is about $42.39$ – Ross Millikan Aug 19 '17 at 1:15
• @RossMillikan instead of asking for where it switches from being below a $50\%$ chance to being above a $50\%$ chance, isn't the OP asking for the expected number of draws until a match occurs? Will those two necessarily be the same? I'm getting a different result (fixed minor typo in equation) getting a result closer to $\approx 45.788$ – JMoravitz Aug 19 '17 at 1:20
• @JMoravitz: I'll reopen it. I don't know an easy approach to the expected number version, but would expect it to be close to the $50\%$ probability number. No, they won't be the same because there is the long tail to high numbers. – Ross Millikan Aug 19 '17 at 2:10
• @RossMillikan well, the pdf is straightforward (and is included in one of your two links in some form) and we can apply the definition of expected value. I don't know of a clean way to simplify the sum by hand, but computers can calculate it easily enough for us. – JMoravitz Aug 19 '17 at 2:12 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9869795095031688,
"lm_q1q2_score": 0.899315694924836,
"lm_q2_score": 0.9111797015700341,
"openwebmath_perplexity": 279.07212059651675,
"openwebmath_score": 0.7660864591598511,
"tags": null,
"url": "https://math.stackexchange.com/questions/2398684/average-number-of-selections-before-duplicate-picked"
} |
It is impossible to have gotten a duplicate on the first draw. It is impossible to have not gotten a duplicate by the 1297'th draw by pigeon-hole principle.
To have gotten your first duplicate on the $k$'th draw, you need the first $k-1$ draws to all be distinct and the $k$'th to be a duplicate.
The first draw will always be distinct. The second will be distinct from the first with probability $\frac{1295}{1296}$. The third will be distinct from the first two with probability $\frac{1294}{1296}$ and so on... the $(n)$'th will be distinct from the earlier $n-1$ with probability $\frac{1296-n+1}{1296}$. Multiplying these, we get for $n$ draws to all be distinct, this will occur with probability $\frac{1296\frac{n}{~}}{1296^n}$ where $x\frac{n}{~}$ represents a falling factorial $x\frac{n}{~}=\underbrace{x(x-1)(x-2)\cdots (x-n+1)}_{n~\text{terms in the product}}=\frac{x!}{(x-n)!}$.
Next, supposing $k-1$ distinct values have all been taken, for the $k$'th to duplicate one of the earlier results, this will occur with probability $\frac{k-1}{1296}$
We have then the probability distribution function for $X$, the number of draws until the first duplicate:
$$Pr(X=k)=\frac{(k-1)1296\frac{k-1}{~}}{1296^k}$$
Applying the definition of expected value for a pdf: $E[X]=\sum\limits_{k\in\Delta} kPr(X=k)$ we have then the expected value is
$$\sum\limits_{k=2}^{1297}\frac{k(k-1)1296\frac{k-1}{~}}{1296^k}\approx 45.7889$$
• As a minor aside, I was having difficulty with the link. Trying to write it as [linkname](actuallinkgoeshere) with parenthesis appearing in the link part of it was being chopped off, and replacing parenthesis with brackets was causing the calculation to time out. Tinyurl's aren't permenant... does anyone have a suggestion other than just hiding it with a spoiler tag like I have? – JMoravitz Aug 19 '17 at 2:35 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9869795095031688,
"lm_q1q2_score": 0.899315694924836,
"lm_q2_score": 0.9111797015700341,
"openwebmath_perplexity": 279.07212059651675,
"openwebmath_score": 0.7660864591598511,
"tags": null,
"url": "https://math.stackexchange.com/questions/2398684/average-number-of-selections-before-duplicate-picked"
} |
With $n$ objects the expected time until the first repeat is exactly $$\mathbb{E}(T)=\int_0^\infty \left(1+{x\over n}\right)^ne^{-x}\,dx,$$ and approximately equal to $\sqrt{n\pi/2}.$
You can find a derivation of this formula at my answer here: Variance of time to find first duplicate
For $n=1296$ the exact formula gives $\mathbb{E}(T)\approx 45.78885405,$ while the approximation gives $\sqrt{1296\pi/2}\approx 45.11930893.$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9869795095031688,
"lm_q1q2_score": 0.899315694924836,
"lm_q2_score": 0.9111797015700341,
"openwebmath_perplexity": 279.07212059651675,
"openwebmath_score": 0.7660864591598511,
"tags": null,
"url": "https://math.stackexchange.com/questions/2398684/average-number-of-selections-before-duplicate-picked"
} |
# Find relationship between $a, b, c, f, g, h$
Given that - $$a=a_1 a_2$$ $$b=b_1 b_2$$ $$c=c_1 c_2$$ $$h=a_2 b_1 + b_2 a_1$$ $$g=a_1 c_2 + a_2 c_1$$ $$f=b_1 c_2 + b_2 c_1$$
Find the relationship between $a, b, c, f, g, h$
My Attempt: I could not see how I could exploit the symmetry of the equations to directly get an answer, so I tried to solve them as 6 simultaneous equations by - $$a_2=\frac{a}{a_1}$$ $$b_2=\frac{b}{b_1}$$ $$c_2=\frac{c}{c_1}$$ and put these values into the remaining 3 equations to obtain - $$a_1 b_1 h = a {b_1}^2 + b {a_1}^2$$ $$a_1 c_1 g = a {c_1}^2 + c {a_1}^2$$ $$c_1 b_1 f = c {b_1}^2 + b {c_1}^2$$ Then, I treated the last 2 equations as quadratics in $a_1$ and $b_1$; found their values using the quadratic formula and input those into the first of the last 3 equations above.
Then I found the value of $c_1$ using that and then the value of the remaining - $a_1, a_2, b_1, b_2, c_2$.
Then, when I finally input these values into any of the equations I got a tautology (which, as I now realize - too late - was doomed to happen from the very beginning, due to my approach).
============================================================
So, How should I go about finding the relation between $a, b, c, f, g, h$?
OR,
Equally - How do I eliminate $a_1, a_2, b_1, b_2, c_1, c_2$ from the equations?
I just need a hint on how I could exploit the symmetry of the equations. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232924970204,
"lm_q1q2_score": 0.8991859751661603,
"lm_q2_score": 0.9149009613738741,
"openwebmath_perplexity": 294.01646413254554,
"openwebmath_score": 0.8017062544822693,
"tags": null,
"url": "https://math.stackexchange.com/questions/2287316/find-relationship-between-a-b-c-f-g-h"
} |
I just need a hint on how I could exploit the symmetry of the equations.
• What does "Find the relationship between a,b,c,f,g,h" mean exactly? One could argue that the initial six equalities already gives a relationship among them. Are you looking for a polynomial expression in a,b,c,f,g,h (and not involving the subscripted variables) that equals 0? – Greg Martin May 19 '17 at 3:36
• Hint: consider $af^2+bg^2+ch^2$. – Greg Martin May 19 '17 at 3:38
• @GregMartin Yes, you're right I need a single expression in a, b, c, f, g, h without the subscripted variables. As an example of such an expression - $\frac{c^{2}}{2a}=\frac{f^{3}}{5g^{2}} +h tan(\frac{b}{a})$ The expression could include logs and trig functions (although I don't think they will be necessary) – Quantum Sphinx May 19 '17 at 3:39
• @GregMartin Yes, that is the kind of expression I need – Quantum Sphinx May 19 '17 at 3:43
Notice that $a,b,c,h,g,f$ are represented by staggered multiplication of $a_1,a_2,b_1,b_2,c_1,c_2$, and one way to align them is to multiply them together, and then we could rearrange how they are combined together. So symmetry is the key here.
$$h\cdot g \cdot f=(a_2 b_1 + b_2 a_1)(a_1 c_2 + a_2 c_1)(b_1 c_2 + b_2 c_1)$$ $$=a_2b_1a_1c_2b_1c_2 + a_2b_1a_1c_2b_2c_1+a_2b_1a_2c_1b_1c_2 + a_2b_1a_2c_1b_2c_1$$$$+b_2a_1a_1c_2b_1c_2+b_2a_1a_1c_2b_2c_1+b_2a_1a_2c_1b_1c_2+b_2a_1a_2c_1b_2c_1$$
$$=ab_1^2c_2^2+abc+a_2^2b_1^2c+a_2^2c_1^2b+a_1^2c_2^2b+a_1^2b_2^2c+abc+b_2^2c_1^2a$$ $$=2abc + a(b_1^2c_2^2+b_2^2c_1^2)+b(a_1^2c_2^2+a_2^2c_1^2) + c(a_2^2b_1^2+a_1^2b_2^2)$$ $$=2abc+a((b_1c_2 + b_2c_1)^2-2b_1c_2b_2c_1)+b((a_1c_2+a_2c_1)^2-2a_1c_2a_2c_1)+c((a_2b_1+a_1b_2)^2-2a_2b_1a_1b_2)$$ Thus $$hgf=2abc + a(f^2-2bc)+b(g^2-2ac)+c(h^2-2ab)$$ $$hgf +4abc -af^2-bg^2-ch^2=0$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232924970204,
"lm_q1q2_score": 0.8991859751661603,
"lm_q2_score": 0.9149009613738741,
"openwebmath_perplexity": 294.01646413254554,
"openwebmath_score": 0.8017062544822693,
"tags": null,
"url": "https://math.stackexchange.com/questions/2287316/find-relationship-between-a-b-c-f-g-h"
} |
# Find the Numbers
Status
Not open for further replies.
##### Full Member
There are two numbers whose sum is 53. Three times the smaller number is equal to 19 more than the larger number. What are the numbers?
Set up:
Let x = large number
Let y = small number
x + y = 53...Equation A
3y = x + 19....Equation B
x + y = 53
y = 53 - x...Plug into B.
3(53 - x) = x + 19
159 - 3x = x + 19
-3x - x = 19 - 159
-4x = -140
x = -140/-4
x = 35...Plug into A or B.
I will use A.
35 + y = 53
y = 53 - 35
y = 18.
The numbers are 18 and 35.
Yes?
#### JeffM
##### Elite Member
Do the numbers satisfy both equation?
$$\displaystyle 35 + 18 = 53.$$ Checks.
$$\displaystyle 3 * 18 = 54 = 35 + 19.$$ Checks.
In algebra, you can always check your own MECHANICAL work, and you should. It avoids mistakes, builds confidence, is a necessary skill for taking tests, and, most importantly, is what you will need in any job that expects you to be able to do math.
#### Subhotosh Khan
##### Super Moderator
Staff member
There are two numbers whose sum is 53. Three times the smaller number is equal to 19 more than the larger number. What are the numbers?
Set up:
Let x = large number
Let y = small number
x + y = 53...Equation A
3y = x + 19....Equation B
x + y = 53
y = 53 - x...Plug into B.
3(53 - x) = x + 19
159 - 3x = x + 19
-3x - x = 19 - 159
-4x = -140
x = -140/-4
x = 35...Plug into A or B.
I will use A.
35 + y = 53
y = 53 - 35
y = 18.
The numbers are 18 and 35.
Yes?
When possible check your work. Most of the time that is a part of the process of solution.
There is a shorter way to accomplish the algebra/arithmetic part.
You have two equations,
x + y = 53...Equation A
3y = x + 19....Equation B
rewrite B to collect all the unknowns to LHS
x + y = 53...Equation A
3y - x = 19....Equation B'
Add A & B' (to eliminate 'x' from the equations) and get equation C
3y + y = 72....Equation C
4y = 72
y = 18
Use this value in equation 'A'
x + 18 = 53...Equation A | {
"domain": "freemathhelp.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232879690035,
"lm_q1q2_score": 0.8991859630418294,
"lm_q2_score": 0.9149009532527358,
"openwebmath_perplexity": 1169.4730670751283,
"openwebmath_score": 0.5126290917396545,
"tags": null,
"url": "https://www.freemathhelp.com/forum/threads/find-the-numbers.115989/"
} |
3y + y = 72....Equation C
4y = 72
y = 18
Use this value in equation 'A'
x + 18 = 53...Equation A
x = 53- 18 = 35
##### Full Member
When possible check your work. Most of the time that is a part of the process of solution.
There is a shorter way to accomplish the algebra/arithmetic part.
You have two equations,
x + y = 53...Equation A
3y = x + 19....Equation B
rewrite B to collect all the unknowns to LHS
x + y = 53...Equation A
3y - x = 19....Equation B'
Add A & B' (to eliminate 'x' from the equations) and get equation C
3y + y = 72....Equation C
4y = 72
y = 18
Use this value in equation 'A'
x + 18 = 53...Equation A
x = 53- 18 = 35
What is wrong with my method?
#### Dr.Peterson
##### Elite Member
Nothing is wrong with your method. You used substitution, and did it correctly; Khan used addition, which can take just a little less writing than what you did, but is certainly not the only correct way, or even necessarily "better".
##### Full Member
Nothing is wrong with your method. You used substitution, and did it correctly; Khan used addition, which can take just a little less writing than what you did, but is certainly not the only correct way, or even necessarily "better".
There are several methods for solving two equations in two variables, right? Matrix algebra is another useful tool.
#### Dr.Peterson
##### Elite Member
Correct.
In fact, each method can be applied to a given system of equations in several ways (which makes it interesting to grade tests). You can solve either equation for either variable and substitute, or eliminate either variable from the equations by adding, then get the other variable in a couple ways. And you can solve the matrix form by several different techniques. When there are three or more variables, it gets even better!
But still, solving the equations is the "easy" (routine) part, compared to setting them up from a word problem.
##### Full Member
Correct. | {
"domain": "freemathhelp.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232879690035,
"lm_q1q2_score": 0.8991859630418294,
"lm_q2_score": 0.9149009532527358,
"openwebmath_perplexity": 1169.4730670751283,
"openwebmath_score": 0.5126290917396545,
"tags": null,
"url": "https://www.freemathhelp.com/forum/threads/find-the-numbers.115989/"
} |
##### Full Member
Correct.
In fact, each method can be applied to a given system of equations in several ways (which makes it interesting to grade tests). You can solve either equation for either variable and substitute, or eliminate either variable from the equations by adding, then get the other variable in a couple ways. And you can solve the matrix form by several different techniques. When there are three or more variables, it gets even better!
But still, solving the equations is the "easy" (routine) part, compared to setting them up from a word problem.
We can also graph two equations to see where they cross each other. The crossing point is the solution in the form (x, y).
#### Jomo
##### Elite Member
I know that you can check these problems. Just admit that you like posting here. | {
"domain": "freemathhelp.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232879690035,
"lm_q1q2_score": 0.8991859630418294,
"lm_q2_score": 0.9149009532527358,
"openwebmath_perplexity": 1169.4730670751283,
"openwebmath_score": 0.5126290917396545,
"tags": null,
"url": "https://www.freemathhelp.com/forum/threads/find-the-numbers.115989/"
} |
Home > standard error > proportion standard error
# Proportion Standard Error
## Contents
repeatedly randomly drawn from a population, and the proportion of successes in each sample is recorded ($$\widehat{p}$$),the distribution of the sample
## Standard Error Of Proportion Formula
proportions (i.e., the sampling distirbution) can be approximated by a normal standard error of proportion definition distribution given that both $$n \times p \geq 10$$ and $$n \times (1-p) \geq 10$$. This is sample proportion formula known as theRule of Sample Proportions. Note that some textbooks use a minimum of 15 instead of 10.The mean of the distribution of sample proportions is equal | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
to the population proportion ($$p$$). The standard deviation of the distribution of sample proportions is symbolized by $$SE(\widehat{p})$$ and equals $$\sqrt{\frac {p(1-p)}{n}}$$; this is known as thestandard error of $$\widehat{p}$$. The symbol $$\sigma _{\widehat p}$$ is also used to signify the standard deviation of the distirbution of sample proportions. Standard Error of the Sample Proportion$## Sample Proportion Calculator SE(\widehat{p})= \sqrt{\frac {p(1-p)}{n}}$If $$p$$ is unknown, estimate $$p$$ using $$\widehat{p}$$The box below summarizes the rule of sample proportions: Characteristics of the Distribution of Sample ProportionsGiven both $$n \times p \geq 10$$ and $$n \times (1-p) \geq 10$$, the distribution of sample proportions will be approximately normally distributed with a mean of $$\mu_{\widehat{p}}$$ and standard deviation of $$SE(\widehat{p})$$Mean $$\mu_{\widehat{p}}=p$$Standard Deviation ("Standard Error")$$SE(\widehat{p})= \sqrt{\frac {p(1-p)}{n}}$$ 6.2.1 - Marijuana Example 6.2.2 - Video: Pennsylvania Residency Example 6.2.3 - Military Example ‹ 6.1.2 - Video: Two-Tailed Example, StatKey up 6.2.1 - Marijuana Example › Printer-friendly version Navigation Start Here! Welcome to STAT 200! Search Course Materials Faculty login (PSU Access Account) Lessons Lesson 0: Statistics: The “Big Picture” Lesson 1: Gathering Data Lesson 2: Turning Data Into Information Lesson 3: Probability - 1 Variable Lesson 4: Probability - 2 Variables Lesson 5: Probability Distributions Lesson 6: Sampling Distributions6.1 - Simulation of a Sampling Distribution of a Proportion (Exact Method) 6.2 - Rule of Sample Proportions (Normal Approximation Method)6.2 | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
0 otherwise. The standard deviation of any variable involves the https://onlinecourses.science.psu.edu/stat200/node/43 expression . Let's suppose there are m 1s (and n-m 0s) among the n subjects. Then, and is equal to (1-m/n) for m observations and 0-m/n http://www.jerrydallal.com/lhsp/psd.htm for (n-m) observations. When these results are combined, the final result is and the sample variance (square of the SD) of the 0/1 observations is The sample proportion is the mean of n of these observations, so the standard error of the proportion is calculated like the standard error of the mean, that is, the SD of one of them divided by the square root of the sample size or Copyright © 1998 Gerard E. Dallal | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
Tables Constants Calendars Theorems Standard Error of Sample Proportion Calculator https://www.easycalculation.com/statistics/standard-error-sample-proportion.php Calculator Formula Download Script Online statistic calculator allows you to estimate the accuracy of the standard error of the sample proportion in the binomial standard deviation. Calculate SE Sample Proportion of Standard standard error Deviation Proportion of successes (p)= (0.0 to 1.0) Number of observations (n)= Binomial SE of Sample proportion= Code to add this calci to your website Just copy and paste the below code to your webpage where you standard error of want to display this calculator. Formula Used: SEp = sqrt [ p ( 1 - p) / n] where, p is Proportion of successes in the sample,n is Number of observations in the sample. Calculation of Standard Error in binomial standard deviation is made easier here using this online calculator. Related Calculators: Vector Cross Product Mean Median Mode Calculator Standard Deviation Calculator Geometric Mean Calculator Grouped Data Arithmetic Mean Calculators and Converters ↳ Calculators ↳ Statistics ↳ Data Analysis Top Calculators Standard Deviation Mortgage Logarithm FFMI Popular Calculators Derivative Calculator Inverse of Matrix Calculator Compound Interest Calculator Pregnancy Calculator Online Top Categories AlgebraAnalyticalDate DayFinanceHealthMortgageNumbersPhysicsStatistics More For anything contact support@easycalculation.com
### Related content
2 x standard error | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
### Related content
2 x standard error
X Standard Error table id toc tbody tr td div id toctitle Contents div ul li a href Times Standard Error a li li a href Computing Standard Error a li ul td tr tbody table p proportion of samples that would fall between and standard deviations above relatedl and below the actual value The standard error SE x standard error is the standard deviation of the sampling distribution of a statistic standard error of x and y most commonly of the mean The term may also be used to refer to an estimate of that standard error of x
2 times standard error mean
Times Standard Error Mean table id toc tbody tr td div id toctitle Contents div ul li a href Standard Error Meaning And Interpretation a li li a href Standard Error Of Means Equation a li li a href Meaning Of Standard Error Bars a li ul td tr tbody table p proportion of samples that would fall between and standard deviations above relatedl and below the actual value The standard error SE standard error of two means is the standard deviation of the sampling distribution of a statistic standard error of two means calculator most commonly of the mean
2 standard error
Standard Error table id toc tbody tr td div id toctitle Contents div ul li a href Standard Error Of The Mean a li li a href What Is A Good Standard Error a li li a href Define Standard Error Of The Mean a li li a href What Does A High Standard Error Mean a li ul td tr tbody table p underestimate the mean by some amount But what's interesting is that the distribution of all these sample relatedl means will itself be normally distributed even if the p h id Standard Error Of The Mean p
2 sample standard error | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
2 sample standard error
Sample Standard Error table id toc tbody tr td div id toctitle Contents div ul li a href Sample Standard Error Equation a li li a href Sample Standard Error Of The Mean Formula a li ul td tr tbody table p randomly sample standard error calculator drawn from the same normally distributed source population belongs to sample standard error excel a normally distributed sampling distribution whose overall mean is equal to zero and whose standard deviation standard sample size standard error error is equal to square root sd na sd nb where sd the variance of the source population
2 sample standard error formula
Sample Standard Error Formula table id toc tbody tr td div id toctitle Contents div ul li a href Standard Error Formula Statistics a li li a href Standard Error Of Estimate Formula a li li a href Percent Error Formula a li ul td tr tbody table p randomly p h id Standard Error Formula Statistics p drawn from the same normally distributed source population belongs to standard error formula proportion a normally distributed sampling distribution whose overall mean is equal to zero and whose standard deviation standard standard error formula regression error is equal to square root sd
2007 excel standard error
Excel Standard Error table id toc tbody tr td div id toctitle Contents div ul li a href Standard Error Function In Excel a li li a href Standard Error Of Mean Excel a li ul td tr tbody table p One relatedl games Xbox games PC calculate standard error excel games Windows games Windows phone games Entertainment All excel standard deviation Entertainment Movies TV Music Business Education Business Students standard error bars in excel educators Developers Sale Sale Find a store Gift cards Products Software services Windows Office Free downloads security standard error excel formula Internet Explorer Microsoft Edge
29 the multiple standard error of estimate is | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
29 the multiple standard error of estimate is
The Multiple Standard Error Of Estimate Is table id toc tbody tr td div id toctitle Contents div ul li a href What Does The Multiple Standard Error Of Estimate Measure a li li a href Standard Error Of Estimate Multiple Regression a li ul td tr tbody table p is used to predict a single dependent variable Y The predicted value of Y is a linear transformation of the X variables such that the sum of squared deviations of the observed and predicted Y is a minimum The computations are more complex however because the interrelationships among all the
2sls standard error
sls Standard Error table id toc tbody tr td div id toctitle Contents div ul li a href sls Standard Error Correction a li li a href Correcting Standard Errors In Two-stage a li li a href Two Stage Least Squares Standard Errors a li li a href sls Regression a li ul td tr tbody table p Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss relatedl the workings and policies of this site About Us Learn p h id sls Standard Error Correction p more
2sls standard error correction
sls Standard Error Correction table id toc tbody tr td div id toctitle Contents div ul li a href Two Stage Least Squares Standard Errors a li li a href sls Estimator a li li a href How To Correct Standard Errors Of A sls Performed By Hand a li ul td tr tbody table p standard errors of a sls performed by hand Date Fri Feb relatedl - No need to do correcting standard errors in two-stage any bootstrapping if it is just linear regression how to difference between iv and sls fix SLS estimates done 'by hand' sysuse
90 confidence standard error | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
90 confidence standard error
Confidence Standard Error table id toc tbody tr td div id toctitle Contents div ul li a href Standard Error Of Measurement Confidence Interval a li li a href Margin Of Error Confidence Interval a li li a href Confidence Standard Deviation a li ul td tr tbody table p we that the true population average is in the shaded area We are relatedl confident This is the level of confidence How many standard confidence interval standard deviation errors away from the mean must we go to be confident From standard error confidence interval calculator -z to z there is
abbreviation standard error mean
Abbreviation Standard Error Mean table id toc tbody tr td div id toctitle Contents div ul li a href Abbreviation For Standard Deviation a li li a href Standard Error Of The Mean Definition a li li a href Scheduled Abbreviation a li li a href Abbreviate Scholarship a li ul td tr tbody table p operator precedence abs x x absolute value of x without regard to sign alpha significance level relatedl of a hypothesis test also type I error rate -a is standard error abbreviation excel the level of the confidence interval ANOVA analysis of variance beta p
abbreviation for standard error in excel
Abbreviation For Standard Error In Excel table id toc tbody tr td div id toctitle Contents div ul li a href Excel Abbreviation For Principal a li li a href Excel Abbreviation Formula a li li a href Standard Error Bars Excel a li ul td tr tbody table p One relatedl games Xbox games PC how to calculate standard error in excel games Windows games Windows phone games Entertainment All excel abbreviation for subtraction Entertainment Movies TV Music Business Education Business Students p h id Excel Abbreviation For Principal p educators Developers Sale Sale Find a store Gift cards
acceptable standard error range | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
acceptable standard error range
Acceptable Standard Error Range table id toc tbody tr td div id toctitle Contents div ul li a href Acceptable Standard Error Values a li li a href What Is A Good Standard Error a li li a href Two Standard Errors a li li a href What If Standard Error Is Zero a li ul td tr tbody table p Ana-Maria imundi Editor-in-ChiefDepartment of Medical Laboratory DiagnosticsUniversity Hospital Sveti Duh Sveti relatedl Duh Zagreb CroatiaPhone p h id Acceptable Standard Error Values p - e-mail address editorial office at biochemia-medica dot com Useful links Events standard error of the
acceptable standard error values
Acceptable Standard Error Values table id toc tbody tr td div id toctitle Contents div ul li a href What Is A Good Standard Error a li li a href How To Interpret Standard Error Of The Mean a li li a href Normal Standard Error a li li a href Standard Error Purpose a li ul td tr tbody table p it comes to determining how well a linear model fits the data However I've stated previously that R-squared is overrated Is there a different goodness-of-fit statistic that can be more relatedl helpful You bet Today I rsquo ll
actual between data error line
Actual Between Data Error Line table id toc tbody tr td div id toctitle Contents div ul li a href Standard Error Of The Regression a li li a href Linear Regression Standard Error a li li a href Standard Error Of Prediction a li ul td tr tbody table p spread of the y values around that average To do this we use the root-mean-square error r m s error To construct relatedl the r m s error you first need to determine the residuals standard error of estimate formula Residuals are the difference between the actual values and
add standard error bars excel 2003 | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
add standard error bars excel 2003
Add Standard Error Bars Excel table id toc tbody tr td div id toctitle Contents div ul li a href How To Add Standard Error Bars In Excel Mac a li li a href How To Add Standard Error Bars In Excel a li li a href How To Add Standard Error Bars In Excel Column Graph a li ul td tr tbody table p Error Bars to Charts and Graphs in Excel TeachExcel SubscribeSubscribedUnsubscribe K Loading Loading Working Add to Want to watch this again later Sign in to add this video to a playlist Sign in relatedl Share
add standard error bars excel 2008
Add Standard Error Bars Excel table id toc tbody tr td div id toctitle Contents div ul li a href How To Add Standard Error Bars In Excel Mac a li li a href How To Add Standard Error Bars In Excel a li li a href How To Add Standard Error Bars In Excel Column Graph a li ul td tr tbody table p error bars in Mac Excel Mason Posner SubscribeSubscribedUnsubscribe Loading Loading Working Add to Want to watch this again later Sign in to add this relatedl video to a playlist Sign in Share More Report how
adding standard error bars in excel 2003
Adding Standard Error Bars In Excel table id toc tbody tr td div id toctitle Contents div ul li a href How To Add Standard Error Bars In Excel a li li a href How To Add Standard Error Bars In Excel Mac a li li a href How To Add Standard Error Bars In Excel Mac a li ul td tr tbody table p Bars to Charts and Graphs in Excel TeachExcel SubscribeSubscribedUnsubscribe K Loading Loading Working Add to Want to watch this again later Sign in to add this video to a playlist Sign in Share More relatedl
anova standard error | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
anova standard error
Anova Standard Error table id toc tbody tr td div id toctitle Contents div ul li a href Anova Standard Deviation a li li a href Anova Confidence Interval a li ul td tr tbody table p women with low daily calcium intakes mg assigned at random to one of three treatments--placebo calcium carbonate calcium citrate maleate Class relatedl Levels Values GROUP CC CCM P Dependent Variable DBMD Sum regression standard error of Source DF Squares Mean Square F Value Pr F Model regression analysis standard error Error Corrected Total R-Square Coeff Var Root MSE DBMD Mean - - p
anova standard error formula
Anova Standard Error Formula table id toc tbody tr td div id toctitle Contents div ul li a href Standard Error In Anova Table a li li a href Standard Error Formula Excel a li li a href Standard Error Formula Proportion a li ul td tr tbody table p Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack relatedl Overflow the company Business Learn more about hiring developers or posting ads with anova
anova standard error calculation
Anova Standard Error Calculation table id toc tbody tr td div id toctitle Contents div ul li a href Standard Error Calculation In Regression a li li a href Standard Error Calculation In R a li ul td tr tbody table p women with low daily calcium intakes mg assigned at random to one of three treatments--placebo calcium carbonate calcium citrate maleate Class Levels Values GROUP relatedl CC CCM P Dependent Variable DBMD Sum of Source DF anova standard error of the mean Squares Mean Square F Value Pr F Model Error anova standard error of estimate Corrected Total R-Square
anova standard error mean | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
anova standard error mean
Anova Standard Error Mean table id toc tbody tr td div id toctitle Contents div ul li a href Standard Error Anova Formula a li li a href Standard Error In Anova Table a li li a href Anova T Test a li ul td tr tbody table p women with low daily calcium intakes mg assigned at relatedl random to one of three treatments--placebo calcium carbonate calcium anova with mean and standard deviation citrate maleate Class Levels Values GROUP CC CCM P Dependent anova standard error of estimate Variable DBMD Sum of Source DF Squares Mean Square F Value
anova standard error regression
Anova Standard Error Regression table id toc tbody tr td div id toctitle Contents div ul li a href Standard Error Of Regression Coefficient a li li a href Standard Error Of Regression Stata a li ul td tr tbody table p to predict muscle strength Model Summary b R R Square Adjusted R Square Std Error of the Estimate a a Predictors Constant LBM b Dependent relatedl Variable STRENGTH FONT ANOVA Source Sum of Squares df Mean Square anova standard error of the mean F Sig Regression Residual Total anova standard error of estimate Coefficients Variable Unstandardized Coefficients Standardized
anova standard error estimate
Anova Standard Error Estimate table id toc tbody tr td div id toctitle Contents div ul li a href Anova Standard Error Of The Mean a li li a href Standard Error Of Estimate Calculator a li li a href Standard Error Of Estimate Excel a li ul td tr tbody table p Tour Start here for a quick overview of the site Help Center Detailed answers to relatedl any questions you might have Meta Discuss the workings standard error of estimate anova table and policies of this site About Us Learn more about Stack Overflow p h id Anova
ap statistics standard error | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
ap statistics standard error
Ap Statistics Standard Error table id toc tbody tr td div id toctitle Contents div ul li a href Ap Stats Standard Deviation a li li a href Standard Error Formula a li li a href Standard Error Definition a li ul td tr tbody table p test AP formulas FAQ relatedl AP study guides AP calculators Binomial Chi-square ap stats standard error f Dist Hypergeometric Multinomial Negative binomial Normal Poisson how long is ap statistics exam t Dist Random numbers Probability Bayes rule Combinations permutations Factorial Event counter Wizard Graphing p h id Ap Stats Standard Deviation p Scientific
ap statistics standard error formula
Ap Statistics Standard Error Formula table id toc tbody tr td div id toctitle Contents div ul li a href Standard Error Formula Proportion a li li a href Standard Error Formula Regression a li li a href Standard Error Of Estimate Formula a li ul td tr tbody table p test AP formulas FAQ AP study guides AP calculators Binomial Chi-square f Dist Hypergeometric Multinomial Negative binomial Normal Poisson t Dist Random numbers Probability Bayes rule Combinations permutations Factorial Event counter Wizard relatedl Graphing Scientific Financial Calculator books AP calculator review Statistics AP study standard error formula excel guides
application of standard error
Application Of Standard Error table id toc tbody tr td div id toctitle Contents div ul li a href Small Standard Error a li li a href What Is Standard Error Used For In Statistics a li ul td tr tbody table p proportion of samples that would fall between and standard deviations above and below relatedl the actual value The standard error SE is sample standard error the standard deviation of the sampling distribution of a statistic most commonly standard error statistics of the mean The term may also be used to refer to an estimate of that standard
appropriate use of standard error | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
appropriate use of standard error
Appropriate Use Of Standard Error table id toc tbody tr td div id toctitle Contents div ul li a href When To Use Standard Error Of The Mean a li li a href When To Use Standard Error Versus Standard Deviation a li li a href When To Use Standard Error Bars a li ul td tr tbody table p error of the mean Is the choice between these down to personal preference or is one favoured relatedl in the scientific field over another Topics Standard Deviation use standard error standard deviation Questions Followers Follow Standard Error Questions use standard
are standard error and standard deviation the same
Are Standard Error And Standard Deviation The Same table id toc tbody tr td div id toctitle Contents div ul li a href Standard Deviation And Standard Error Are Similar Except For The Following a li li a href What Does Standard Error Measure a li ul td tr tbody table p proportion of samples that would fall between and standard relatedl deviations above and below the actual value The are standard error and standard deviation the same thing standard error SE is the standard deviation of the sampling distribution se formula of a statistic most commonly of the mean
are standard error and deviation the same
Are Standard Error And Deviation The Same table id toc tbody tr td div id toctitle Contents div ul li a href Standard Error Deviation Difference a li li a href Standard Error Deviation Formula a li li a href What s The Difference Between Standard Error And Standard Deviation a li ul td tr tbody table p proportion of samples that would fall between and standard deviations above and below the actual relatedl value The standard error SE is the standard deviation are standard error and standard deviation the same thing of the sampling distribution of a statistic most
arrow indicates standard error estimate | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
arrow indicates standard error estimate
Arrow Indicates Standard Error Estimate table id toc tbody tr td div id toctitle Contents div ul li a href Standard Error Regression Estimate a li li a href Multiple Standard Error Of Estimate a li li a href Standard Error Of Estimate Formula a li ul td tr tbody table p a linear equation system and was first developed by Sewall Wright in the s for use in phylogenetic studies Path Analysis was adopted by the social sciences in the s relatedl and has been used with increasing frequency in the ecological standard error of estimate calculator literature since
as n increases the standard error decreases
As N Increases The Standard Error Decreases table id toc tbody tr td div id toctitle Contents div ul li a href As The Sample Size Increases The Standard Error Of The Mean a li li a href As The Sample Size Increases The Standard Error Also Increases a li li a href Why Does The Standard Error Of The Mean Decreases As The Sample Size Increases a li li a href As The Sample Size Increases The Standard Deviation Of The Population Decreases a li ul td tr tbody table p using Windows or NT When asked if you
as standard error of the mean increases
As Standard Error Of The Mean Increases table id toc tbody tr td div id toctitle Contents div ul li a href What Happens To The Standard Error Of The Mean As N Increases a li li a href What Happens To The Standard Deviation As The Mean Increases a li ul td tr tbody table p using Windows or NT When asked if you want to install the sampling control click on Yes relatedl When we draw a sample from a population as the sample size increases the standard error of the mean and calculate a sample statistic such
as sample size increases the standard error of m | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
as sample size increases the standard error of m
As Sample Size Increases The Standard Error Of M table id toc tbody tr td div id toctitle Contents div ul li a href The Larger The Sample Size The Smaller The Standard Error a li li a href One Standard Error Of The Mean a li li a href As Sample Size Increases The Standard Deviation a li ul td tr tbody table p WorkSocial MediaSoftwareProgrammingWeb Design DevelopmentBusinessCareersComputers Online Courses B B Solutions Shop for Books San Francisco CA Brr it s cold outside Search Submit Learn more with dummies Enter your email to join our mailing list for
as sample size increases the standard error tends to
As Sample Size Increases The Standard Error Tends To table id toc tbody tr td div id toctitle Contents div ul li a href As The Size Of A Random Sample Increases The Standard Error Of The Mean Decreases a li li a href As The Sample Size Increases The Standard Error Also Increases a li li a href What Is Standard Error Of The Mean For Dummies a li ul td tr tbody table p WorkSocial MediaSoftwareProgrammingWeb Design DevelopmentBusinessCareersComputers Online Courses B B Solutions Shop for Books San Francisco CA Brr it s cold outside Search Submit Learn more
asp net redirect to standard error page
Asp Net Redirect To Standard Error Page table id toc tbody tr td div id toctitle Contents div ul li a href Redirect Standard Error Unix a li li a href Redirect Standard Error Bash a li ul td tr tbody table p Websites Community Support ASP NET Community Standup ForumsHelp Web Forms Guidance Videos Samples Forum Books relatedl Open Source Older Versions - Getting Started Getting redirect standard error and output to different files StartedGetting Started with ASP NET Web Forms and Visual Studio Getting redirect standard error and output to dev null Started with Web Forms and Visual
asymptotic standard error wiki | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
asymptotic standard error wiki
Asymptotic Standard Error Wiki table id toc tbody tr td div id toctitle Contents div ul li a href Asymptotic Standard Errors Definition a li li a href Asymptotic Definition Statistics a li li a href Asymptotic Synonym a li li a href Asymptotic Analysis a li ul td tr tbody table p standard error and standard error I know about standard error but not getting idea about the asymptotic standard error and how it is related to standard error relatedl Topics Asymptotic Statistics Questions Followers Follow Statistical Physics asymptotic standard error gnuplot Questions Followers Follow Basic Statistics Questions Followers
asymptotic standard error definition
Asymptotic Standard Error Definition table id toc tbody tr td div id toctitle Contents div ul li a href Principle Of Maximum Likelihood a li li a href Asymptotic Standard Error Gnuplot a li ul td tr tbody table p standard error and standard error I know about standard error but not getting idea about the asymptotic standard error and how it is related to standard error Topics Asymptotic Statistics relatedl Questions Followers Follow Statistical Physics Questions Followers meaning of asymptotic standard error Follow Basic Statistics Questions Followers Follow Analytical Statistics Questions define asymptotic standard error Followers Follow Standard Error
asymptotic standard error for me | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
asymptotic standard error for me
Asymptotic Standard Error For Me table id toc tbody tr td div id toctitle Contents div ul li a href Standard Error Formula a li li a href Standard Error Vs Standard Deviation a li ul td tr tbody table p standard error and standard error I know about standard error but not getting idea about the asymptotic standard error and how it is related to standard error Topics Asymptotic relatedl Statistics Questions Followers Follow Statistical Physics asymptotic standard error gnuplot Questions Followers Follow Basic Statistics Questions Followers Follow Analytical Statistics asymptotic standard errors definition Questions Followers Follow Standard Error
asymptotic standard error matlab
Asymptotic Standard Error Matlab table id toc tbody tr td div id toctitle Contents div ul li a href Asymptotic Standard Error Gnuplot a li li a href Calculate Standard Error Matlab a li li a href Plot Standard Error Matlab a li li a href Confidence Interval Matlab a li ul td tr tbody table p toolboxes and p h id Asymptotic Standard Error Gnuplot p other File Exchange content using Add-On Explorer asymptotic standard error definition in MATLAB raquo Watch video Highlights from Toolkit on Econometrics standard error matlab regression and Economics Teaching LOGIT m EM TOBIT m
asymptotic standard error stata
Asymptotic Standard Error Stata table id toc tbody tr td div id toctitle Contents div ul li a href Robust Standard Error Stata a li li a href Cluster Standard Error Stata a li li a href Asymptotic Standard Error Gnuplot a li ul td tr tbody table p on statistics Stata Journal Stata Press Stat Transfer Gift Shop Purchase Order Stata Request a quote Purchasing FAQs relatedl Bookstore Stata Press books Books on Stata Books on standard error regression stata statistics Stat Transfer Stata Journal Gift Shop Training NetCourses Classroom and web On-site standard error stata output Video tutorials
asymptotic standard error sas | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
asymptotic standard error sas
Asymptotic Standard Error Sas table id toc tbody tr td div id toctitle Contents div ul li a href Calculate Standard Error In Sas a li li a href Standard Deviation Sas a li li a href Variance Sas a li li a href T Test Sas a li ul td tr tbody table p row and column variables of the contingency table The following are measures of ordinal association that consider whether the column variable Y tends to increase relatedl as the row variable X increases gamma Kendall s tau- Stuart s tau- standard error sas proc means and
asymptotic standard error equals zero
Asymptotic Standard Error Equals Zero table id toc tbody tr td div id toctitle Contents div ul li a href Standard Error Vs Standard Deviation a li li a href Margin Of Error a li li a href Type Error a li ul td tr tbody table p Upload Documents Write Course Advice Refer your Friends Earn Money Upload Documents Apply for Scholarship Create relatedl Q A pairs Become a Tutor Find Study Resources by standard error formula School by Literature Guides by Subject Get Instant Tutoring Help Ask a Tutor p h id Standard Error Vs Standard Deviation p
asymptotic standard error wikipedia
Asymptotic Standard Error Wikipedia table id toc tbody tr td div id toctitle Contents div ul li a href Asymptotic Standard Error Gnuplot a li li a href Standard Error Regression Wikipedia a li li a href Asymptotic Synonym a li li a href Asymptotic Distribution a li ul td tr tbody table p February Learn how and when to remove this template message It has been suggested that this article be merged into Studentization Discuss Proposed relatedl since May In statistics the t-statistic is a ratio p h id Asymptotic Standard Error Gnuplot p of the departure of an
autocorrelation standard error | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
autocorrelation standard error
Autocorrelation Standard Error table id toc tbody tr td div id toctitle Contents div ul li a href Autocorrelation Variance a li li a href Autocorrelation Standard Error Underestimate a li li a href Robust Standard Errors Autocorrelation a li ul td tr tbody table p Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss relatedl the workings and policies of this site About Us Learn multicollinearity standard error more about Stack Overflow the company Business Learn more about hiring developers or posting autocorrelation confidence interval ads
average standard error
Average Standard Error table id toc tbody tr td div id toctitle Contents div ul li a href Average Standard Deviation a li li a href Average Variance a li li a href Average Coefficient Of Variation a li ul td tr tbody table p Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn relatedl more about Stack Overflow the company Business Learn more about hiring developers mean standard error or posting ads with us Cross Validated
average standard error calculator
Average Standard Error Calculator table id toc tbody tr td div id toctitle Contents div ul li a href Standard Error Of Estimate Calculator a li li a href Standard Error Calculator Two Samples a li li a href Standard Error Calculator For Regression a li ul td tr tbody table p of the mean of a set of numbers Standard p h id Standard Error Calculator Two Samples p Error of the Mean The standard error of the mean is the standard deviation of p h id Standard Error Calculator For Regression p the sample mean estimate of a
average standard error on excel | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
average standard error on excel
Average Standard Error On Excel table id toc tbody tr td div id toctitle Contents div ul li a href Average Standard Error Of The Mean a li li a href Standard Error Of Average Partial Effect a li li a href Standard Error Of Average Formula a li li a href Standard Error Of The Average Of Multiple Measurements a li ul td tr tbody table p the toolbar at the top A menu will appear that relatedl says Paste Function Select Stastical from the left hand side p h id Average Standard Error Of The Mean p of
average standard error arcgis
Average Standard Error Arcgis table id toc tbody tr td div id toctitle Contents div ul li a href Average Standard Error Of The Mean a li li a href Percentage Standard Error a li li a href Cross Validation a li ul td tr tbody table p for Developers Tools to build location-aware apps ArcGIS Solutions Free template relatedl maps and apps for your industry ArcGIS Marketplace standard deviation arcgis Get apps and data for your organization Documentation Pricing Support Esri p h id Average Standard Error Of The Mean p Sign In user My Profile Sign Out Go
average standard error of the mean
Average Standard Error Of The Mean table id toc tbody tr td div id toctitle Contents div ul li a href Maksud Standard Error a li li a href Mean Average Standard Deviation a li li a href Mean Average Standard Deviation Calculator a li ul td tr tbody table p proportion of samples that would fall between and standard deviations above and below the actual value The standard error SE is relatedl the standard deviation of the sampling distribution of a statistic most derive standard error of mean commonly of the mean The term may also be used to
average standard error of measurement | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
average standard error of measurement
Average Standard Error Of Measurement table id toc tbody tr td div id toctitle Contents div ul li a href Standard Error Of Measurement Formula a li li a href Standard Error Of Measurement Example a li li a href Standard Error Of Measurement Vs Standard Error Of Mean a li li a href Standard Error Of Measurement Spss a li ul td tr tbody table p latter is impossible standardized tests usually have an associated standarderror of measurement SEM an index of the expected variation in relatedl observedscores due to measurement error The SEM is in standard standard error
average standard error measurement
Average Standard Error Measurement table id toc tbody tr td div id toctitle Contents div ul li a href Standard Error Of Measurement Formula a li li a href Standard Error Of Measurement And Confidence Interval a li li a href Standard Error Of Measurement Example a li ul td tr tbody table p proportion of samples that would fall between and standard deviations above and below the actual value The standard relatedl error SE is the standard deviation of the sampling distribution standard error of measurement calculator of a statistic most commonly of the mean The term may also
average standard error of prediction
Average Standard Error Of Prediction table id toc tbody tr td div id toctitle Contents div ul li a href Standard Error Of Prediction Linear Regression a li li a href Standard Error Of Prediction Calculator a li li a href Standard Error Of Prediction Interval a li li a href Standard Error Of Prediction Stata a li ul td tr tbody table p the estimate from a scatter plot Compute the standard error of the estimate based on errors of prediction Compute the standard error relatedl using Pearson's correlation Estimate the standard error of the estimate standard error of
bash error message to stderr | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
bash error message to stderr
Bash Error Message To Stderr table id toc tbody tr td div id toctitle Contents div ul li a href How To Redirect Stderr And Stdout To A File a li li a href Bash Standard Error To Variable a li li a href Bash Echo To Standard Error a li li a href Bash Print To Standard Error a li ul td tr tbody table p here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings relatedl and policies of this site About Us Learn more about
bash standard error output
Bash Standard Error Output table id toc tbody tr td div id toctitle Contents div ul li a href Bash Redirect Error Output To dev null a li li a href Bash Standard Error To Variable a li li a href Bash Output To Stderr a li ul td tr tbody table p a stderr redirect stderr to a stdout redirect stderr and stdout to a file redirect stderr and stdout to stdout redirect stderr relatedl and stdout to stderr 'represents' stdout and stderr bash redirect error output to file A little note for seeing this things with the less
bash standard error redirect
Bash Standard Error Redirect table id toc tbody tr td div id toctitle Contents div ul li a href Bash Pipe Standard Error a li li a href Redirect Standard Error Csh a li ul td tr tbody table p Applications A DT DL DIV A P There are always three default files I SPAN A open stdin TT the keyboard stdout TT the screen and relatedl stderr TT error messages output to the screen These and bash redirect standard error dev null any other open files can be redirected Redirection simply means capturing output from a bash redirect standard
bash standard error | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
bash standard error
Bash Standard Error table id toc tbody tr td div id toctitle Contents div ul li a href Bash Redirect To Dev Null a li li a href How To Redirect Stderr And Stdout To A File a li li a href Bash Echo To Standard Error a li ul td tr tbody table p a stderr redirect stderr to a stdout redirect stderr and stdout to a file redirect stderr and stdout to stdout redirect stderr relatedl and stdout to stderr 'represents' stdout and stderr bash standard error to variable A little note for seeing this things with the
basic definition of standard error of measurement
Basic Definition Of Standard Error Of Measurement table id toc tbody tr td div id toctitle Contents div ul li a href Standard Error Of Measurement Calculator a li li a href Standard Error Of Measurement And Confidence Interval a li li a href Standard Error Of Measurement Vs Standard Deviation a li li a href Standard Error Of Measurement Vs Standard Error Of Mean a li ul td tr tbody table p latter is impossible standardized tests usually have an associated standarderror of measurement SEM relatedl an index of the expected variation in observedscores define standard error of measurement
best estimate of standard error
Best Estimate Of Standard Error table id toc tbody tr td div id toctitle Contents div ul li a href Estimate Standard Error Calculator a li li a href Estimate Standard Error Of Proportion a li li a href Estimate Standard Error Bootstrap a li li a href Estimate The Standard Error Of The Sample Mean a li ul td tr tbody table p it comes to determining how well a linear model fits the data However I've stated previously that R-squared is overrated relatedl Is there a different goodness-of-fit statistic that can be more how to estimate standard error
bernoulli error bars | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
bernoulli error bars
Bernoulli Error Bars table id toc tbody tr td div id toctitle Contents div ul li a href Standard Error Binary Distribution a li li a href Standard Deviation Of Dummy Variable a li li a href Binomial Error a li li a href Standard Error Is Used In The Calculation Of Both The Z And T Statistic With The Difference That a li ul td tr tbody table p be calculated for a binary variable In a graph showing the progress over time of the probability to relatedl find a pathogen within plant tissues I'm wondering if standard error
beta standard error excel
Beta Standard Error Excel table id toc tbody tr td div id toctitle Contents div ul li a href Standard Error Of Beta Linear Regression a li li a href Standard Error Of Beta Calculator a li li a href Beta Standard Deviation a li ul td tr tbody table p std error in excel ArteSuaveEconomist SubscribeSubscribedUnsubscribe Loading Loading Working Add to Want to watch this again later Sign in to add this video to a playlist relatedl Sign in Share More Report Need to report the video standard error of beta coefficient Sign in to report inappropriate content Sign
beta standard error formula
Beta Standard Error Formula table id toc tbody tr td div id toctitle Contents div ul li a href Standard Error Formula Statistics a li li a href Standard Error Formula Proportion a li li a href Standard Error Formula Regression a li ul td tr tbody table p Curve Z-table Right of Curve Probability and Statistics Statistics Basics Probability Regression Analysis Critical Values Z-Tables Hypothesis Testing Normal Distributions Definition Word Problems T-Distribution relatedl Non Normal Distribution Chi Square Design of Experiments Multivariate Analysis standard error formula excel Sampling in Statistics Famous Mathematicians and Statisticians Calculators Variance and Standard Deviation
beta standard error | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
beta standard error
Beta Standard Error table id toc tbody tr td div id toctitle Contents div ul li a href Beta Variance a li li a href Standard Error Of Regression Coefficient a li li a href Standard Error Of Beta Estimate a li ul td tr tbody table p faq bull rss Community Log In Sign Up Add New Post Question what does beta and standard error mean in a typical GWAS results months ago by iphoenix bull European Union iphoenix bull relatedl wrote Hi all Since I am new to GWAS and statistics beta standard deviation I find it hard
beta standard error regression
Beta Standard Error Regression table id toc tbody tr td div id toctitle Contents div ul li a href Standard Error Of Beta Linear Regression a li li a href Standard Error Of Beta Coefficient Formula a li ul td tr tbody table p Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings relatedl and policies of this site About Us Learn more about Stack standard error of coefficient in linear regression Overflow the company Business Learn more about hiring developers or posting ads with
beta coefficient divided by standard error
Beta Coefficient Divided By Standard Error table id toc tbody tr td div id toctitle Contents div ul li a href T Statistic Coefficient Divided By Standard Error a li li a href Standard Error Of Beta Coefficient Formula a li li a href Regression Coefficient Interpretation a li ul td tr tbody table p here td Nov -Dec font Walk-in - pm td Dec -Feb font By appt here td Feb -May font Walk-in - pm font a May -May font Walk-in relatedl - pm font a May -Aug font By appt here td For t-stat coefficient divided by
big standard error | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
big standard error
Big Standard Error table id toc tbody tr td div id toctitle Contents div ul li a href Importance Of Standard Error Of The Mean a li li a href Large Standard Error In Multiple Regression a li li a href What Does A Large Standard Error Mean a li ul td tr tbody table p Ana-Maria imundi Editor-in-ChiefDepartment of Medical Laboratory DiagnosticsUniversity Hospital Sveti Duh Sveti Duh relatedl Zagreb CroatiaPhone - e-mail large standard error address editorial office at biochemia-medica dot com Useful links Events Follow p h id Importance Of Standard Error Of The Mean p us on
beta standard error calculation
Beta Standard Error Calculation table id toc tbody tr td div id toctitle Contents div ul li a href Standard Error Calculation Excel a li li a href Standard Error Calculation In Regression a li li a href Standard Error Calculation In R a li li a href Margin Of Error Calculation a li ul td tr tbody table p Tour Start here for a quick overview of the site Help Center Detailed relatedl answers to any questions you might have Meta Discuss p h id Standard Error Calculation Excel p the workings and policies of this site About Us
binary standard error
Binary Standard Error table id toc tbody tr td div id toctitle Contents div ul li a href Standard Deviation Binary a li li a href T Test Binary a li li a href Standard Deviation Of Yes No Data a li ul td tr tbody table p Tour Start here for a quick overview of the site Help Center Detailed answers to any questions relatedl you might have Meta Discuss the workings and policies standard error binary data of this site About Us Learn more about Stack Overflow the company Business p h id Standard Deviation Binary p Learn
binomial distribution error bars | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
binomial distribution error bars
Binomial Distribution Error Bars table id toc tbody tr td div id toctitle Contents div ul li a href Binomial Proportion Confidence Interval a li li a href Standard Error Binary Distribution a li li a href Binomial Error a li ul td tr tbody table p be calculated for a binary variable In a graph showing the progress over time of the probability to find a pathogen within plant tissues I'm wondering if standard deviation relatedl or standard error bars can be added The probability to find standard error binomial distribution the pathogen is obtained dividing the number of
binomial standard error formula
Binomial Standard Error Formula table id toc tbody tr td div id toctitle Contents div ul li a href Binomial Distribution Standard Error a li li a href Binomial Sampling Error a li li a href Standard Error Formula Statistics a li li a href Standard Error Of Estimate Formula a li ul td tr tbody table p Tour Start here for a quick overview of the site Help Center Detailed answers to any questions relatedl you might have Meta Discuss the workings and policies p h id Binomial Distribution Standard Error p of this site About Us Learn more
binomial standard error matlab
Binomial Standard Error Matlab table id toc tbody tr td div id toctitle Contents div ul li a href Binomial Standard Deviation a li li a href Binomial T Test a li li a href Binomial Central Limit Theorem a li ul td tr tbody table p Search All Support Resources Support Documentation MathWorks Search MathWorks com relatedl MathWorks Documentation Support Documentation Toggle navigation Trial binomial distribution standard error Software Product Updates Documentation Home Statistics and Machine Learning Toolbox standard error matlab regression Examples Functions and Other Reference Release Notes PDF Documentation Probability Distributions Discrete Distributions Binomial Distribution binomial sampling
binomial standard error mean | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
binomial standard error mean
Binomial Standard Error Mean table id toc tbody tr td div id toctitle Contents div ul li a href Binomial Standard Error Calculator a li li a href Binomial Sampling Error a li li a href Binomial Probability Standard Deviation a li li a href Binomial Confidence Interval a li ul td tr tbody table p -- it is just scaled by a factor n From the properties of the binomial relatedl distribution its distribution has mean and standard deviation Bias binomial distribution standard error and standard error When the proportion p is used to estimate p h id Binomial
binomial standard error excel
Binomial Standard Error Excel table id toc tbody tr td div id toctitle Contents div ul li a href Standard Error Of Binomial Proportion a li li a href Binomial Sampling Error a li li a href Binomial Confidence Interval a li ul td tr tbody table p BINOMDIST Mean Standard Deviation For Binomial Probability Distribution ExcelIsFun SubscribeSubscribedUnsubscribe K Loading Loading Working Add to Want to watch this relatedl again later Sign in to add this video standard deviation of a binomial distribution in excel to a playlist Sign in Share More Report Need to report the binomial standard error
binomial distribution standard error of the mean
Binomial Distribution Standard Error Of The Mean table id toc tbody tr td div id toctitle Contents div ul li a href Binomial Error a li li a href Standard Error Binary Distribution a li li a href Sample Variance Bernoulli a li ul td tr tbody table p Tour Start here for a quick overview of the site Help Center Detailed answers to any questions relatedl you might have Meta Discuss the workings and policies binomial standard error calculator of this site About Us Learn more about Stack Overflow the company Business standard error of binary variable Learn more
binomial error bars | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
binomial error bars
Binomial Error Bars table id toc tbody tr td div id toctitle Contents div ul li a href What s A Confidence Interval a li li a href Standard Error Binomial Distribution a li li a href Binomial Proportion Confidence Interval a li li a href Bernoulli Standard Deviation a li ul td tr tbody table p Tour Start here for a quick overview of the site Help Center Detailed answers to any questions relatedl you might have Meta Discuss the workings and policies p h id What s A Confidence Interval p of this site About Us Learn more | {
"domain": "winbytes.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9840936068886641,
"lm_q1q2_score": 0.899143230082056,
"lm_q2_score": 0.9136765281148513,
"openwebmath_perplexity": 10404.382160598483,
"openwebmath_score": 0.3121330142021179,
"tags": null,
"url": "http://winbytes.org/help/standard-error/proportion-standard-error.html"
} |
# Spectral Decomposition of A and B.
I was given the following question in my linear algebra course.
Let $A$ be a symmetric matrix, $c >0$, and $B=cA$, find the relationship between the spectral decompositions of $A$ and $B$.
From what I understand. If $A$ is a symmetric matrix, then $A=A^T$. A symmetric matrix has $n$ eigenvalues and there exist $n$ linearly independent eigenvectors (because of orthogonality) even if the eigenvalues are not distinct. Since $B=cA$ and $A=A^T$, then we can conclude that $B=cA^T$, which would imply that $B$ is also symmetric, meaning it also has a linearly independent eigenbasis.
Focusing on $A$, since it has a linearly independent eigenbasis, we have $A = PD_aP^{-1}$ by Spectral decomposition where $P$ is the eigenbasis and $D_a$ is the diagonal matrix of $A$ eigenvalues $\lambda_i$ \begin{array} d D_a & = & \begin{bmatrix} \lambda_1 & & \\ &\ddots&\\ & & \lambda_i \end{bmatrix} \end{array}
Now since $B=cA$, then we have $B=cPD_aP^{-1}$, which can be rewritten as $B = PD_bP^{-1}$, where
\begin{array} d D_b & = & cD_a & =c\begin{bmatrix} \lambda_1 & & \\ &\ddots&\\ & & \lambda_i \end{bmatrix} & = & \begin{bmatrix} c\lambda_1 & & \\ &\ddots&\\ & & c\lambda_i \end{bmatrix} \end{array}
From this I can conclude that $B$ and $A$ actually have the same linearly independent eigenbasis. Furthermore, the eigenvalues of $B$ are a scalar multiple of the eigenvalues of $A$ by a factor of $c$.
Have I fully describe the relationship between $A$ and $B$?
$\dfrac{\lambda}{c} I- A$ isn't invertible if and only if $\lambda I- cA$ isn't invertible.
Hence, $\lambda\in Sp(cA)$ if and only if $\dfrac{\lambda}{c}\in Sp(A)$
Yes, I would say that you have fully described the relationship between $A$ and $B$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9867771782476948,
"lm_q1q2_score": 0.8991313336018317,
"lm_q2_score": 0.9111797003640646,
"openwebmath_perplexity": 98.99838089290482,
"openwebmath_score": 0.9967987537384033,
"tags": null,
"url": "https://math.stackexchange.com/questions/1457701/spectral-decomposition-of-a-and-b"
} |
# Why Binomial Distribution formula includes the “not-happening” probability?
Suppose I have a dice with 6 sides, and I let a random variable $X$ be the number of times I get 3 points when I throw the dice.
So I throw the dice for $10$ times, I want to find the probability of getting 3 points from the dice for $4$ times, ie: $P(X=4)$.
Since the order doesn't matter, there are $\binom{10}{4}=210$ ways and the chance of getting a 3 point is $\frac { 1 }{ 6 }$. Also, because I want to have $4$ of such occurrence, it would be $\frac{1}{6}^4$. So, I could just calculate $P(X=4)=\binom{10}{4}\frac { 1 }{ 6 }^4 =0.027006173\approx 2.7\%$.
But, suppose if I use the Binomial Distribution formula, it would be a little different because it needs to multiply the "not-happening" probability to it. The Binomial Distribution looks like this: $$P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}$$
So if I plug in my values, it would be: $$P(X=4)=\binom{10}{4}(\frac{1}{6})^4(\frac{5}{6})^{6}=0.054=5.4\%$$
Here, $2.1\%$ is lesser than $5.4\%$. What's the difference between the two values? Which is the correct value?
Intuitively, I find the Binomial Distribution may be more accurate since it dictates the situation to consider both the happening and not-happening outcomes. But usually, I thought we just multiply the probabilities of what we want it to happen as long as the events are independent. So the first method sounds quite okay too. Eg: What's the probability to get 2 heads out of 5 flips of a fair coin, I just use $\frac { 1}{ 2} \times \frac { 1}{ 2}$. The not-happening probabilities are not cared of. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.982557516796073,
"lm_q1q2_score": 0.8989428098822946,
"lm_q2_score": 0.9149009544128984,
"openwebmath_perplexity": 221.62111013827325,
"openwebmath_score": 0.8770877122879028,
"tags": null,
"url": "http://math.stackexchange.com/questions/110667/why-binomial-distribution-formula-includes-the-not-happening-probability"
} |
-
To get exactly 4 "3 points", the other 6 throws have to not be "3 points". The other 6 throws not being 3 points is also part of what you want to happen if exactly 4 of the 10 throws are 3 points. So, you need to multiply by $(5/6)^6$. By the way, in your second displayed equation, you should have ${10\choose 4}(1/6)^4(5/6)^6$ – David Mitra Feb 18 '12 at 16:33
oh yea, thanks. It should ${10\choose 4}(1/6)^4(5/6)^6$ in the second line. Updated the equation. If I need to multiply the "not-happening" probability, then suppose if I want to find the probability of getting 2 heads out of 5 flip of a fair coin, it wouldn't it be $\binom{5}{2}{ \left( \frac { 1 }{ 2 } \right) }^{ 2 }{ \left( \frac { 1 }{ 2 } \right) }^{ 3 }=\binom{5}{2}{ \left( \frac { 1 }{ 2 } \right) }^{ 5 }$? But shouldn't it be just $\frac { 1}{ 2} \times \frac { 1}{ 2}$? – xenon Feb 18 '12 at 16:51
No, not just $(1/2)(1/2)$ for exactly two heads. You $need$ the other flips to not be heads; which means you multiply by $(1/2)^3$. – David Mitra Feb 18 '12 at 17:10
@xEnOn: The calculation that gave about $5.4$% is based on the correct analysis, and the calculator work is right. The first calculation, the one that gave about $2.7$%, is based on an incorrect analysis (formula). In addition, there was a calculator mistake. The wrong formula gives an answer of about $16.2$%, not $2.7$%. This is clear if you compare the two expressions. The second multiplies the first by $(5/6)^6$, which is less than $1$. – André Nicolas Feb 18 '12 at 17:18
@xEnOn : Please don't write $\frac 16 ^4$ if you mean $\left(\frac16\right)^4$. The former expression should be used only if you mean $(1^6)/4$. – Michael Hardy Feb 18 '12 at 17:53 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.982557516796073,
"lm_q1q2_score": 0.8989428098822946,
"lm_q2_score": 0.9149009544128984,
"openwebmath_perplexity": 221.62111013827325,
"openwebmath_score": 0.8770877122879028,
"tags": null,
"url": "http://math.stackexchange.com/questions/110667/why-binomial-distribution-formula-includes-the-not-happening-probability"
} |
Your first argument is a bit off. $X=4$ means exactly four of the tosses resulted in $3$-points To get exactly four $3$-points, the other six throws have to not be $3$- points. The other six throws not being $3$-points is also part of what you want to happen if exactly four of the 10 throws are $3$-points.
Look at a particular case: the first four throws are 3-points and there are exactly four 3-points. Then the throw sequence was $$3\text{-pt}\ \ 3\text{-pt}\ \ 3\text{-pt}\ \ 3\text{-pt}\ \ \text{not}3\text{-pt}\ \ \text{not}3\text{-pt}\ \ \text{not}3\text{-pt}\ \ \text{not}3\text{-pt}\ \ \text{not}3\text{-pt}\ \ \text{not}3\text{-pt}\ \$$ The probability of this occurring is $(1/6)^4(5/6)^6$.
So, you need to multiply your proposed answer by $(5/6)^6$. So, the correct answer is what is given by the Binomial distribution: ${10\choose4}(1/6)4(5/6)6$.
In your second argument (of the original post), you are correct assuming that you only drew two cards. If you drew three cards, the probability that exactly two were hearts would be $$\underbrace{{13\over52}{12\over 51}{39\over 50}}_{ \text{two hearts, then non heart}}+ \underbrace{{13\over52}{39\over 51}{12\over 50}}_{ \text{ heart, non heart, heart}}+ \underbrace{{39\over52}{13\over 51}{12\over 50}}_{ \text{ non heart, then two hearts}}$$
To be brief you do care about the "not happening" probabilities when you consider the probability that an event happens exactly $n$ times. Knowing that something does not happen is concrete information. In the die example, "not 3-pt" means that six of the flips were 1,2,4,5, or 6 points.
In your coin example, if you want the probability of exactly two heads in five flips, then the three non-head flips must be tails. What is the probability of flipping $H\ H\ T\ T\ T$? It's not $1/4$... It is $1/32$, as you should be able to see from the multiplication rule for a sequence of independent events. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.982557516796073,
"lm_q1q2_score": 0.8989428098822946,
"lm_q2_score": 0.9149009544128984,
"openwebmath_perplexity": 221.62111013827325,
"openwebmath_score": 0.8770877122879028,
"tags": null,
"url": "http://math.stackexchange.com/questions/110667/why-binomial-distribution-formula-includes-the-not-happening-probability"
} |
The number of ways in which you can have exactly two heads in five flips is ${5\choose2}=10$ (it's the number of ways to choose two slots from five in which to put the two heads in; note the other three slots must be tails). The probability of exactly two heads in five flips is ${5\choose2}\cdot(1/2)^2(1/2)^3={10\over 32}={5\over16}$.
To see more concretely why your argument fails, consider this simple case. Toss a fair coin three times. What is the probability of having exactly one head? By your reasoning it is $3\cdot(1/2)^1>1$; which is nonsense. Perhaps it's just $1/2$? This is incorrect also:
The equally likely outcomes here are: $$HHH \ \ \ HHT \ \ \ HTT\ \ \ HTH$$ $$THH \ \ \ THT \ \ \ TTT\ \ \ TTH$$ And we see the probability of exactly one head is ${3\over8}$, which is exactly what the Binomial formuls gives: ${3\over8}={3\choose2}(1/2)^1(1/2)^2$.
Incidentally, you're proposed method would not even give the probability of having at least one head. Here, that probability is $7/8$.
- | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.982557516796073,
"lm_q1q2_score": 0.8989428098822946,
"lm_q2_score": 0.9149009544128984,
"openwebmath_perplexity": 221.62111013827325,
"openwebmath_score": 0.8770877122879028,
"tags": null,
"url": "http://math.stackexchange.com/questions/110667/why-binomial-distribution-formula-includes-the-not-happening-probability"
} |
# Homework Help: Give a big-O estimate of the product of the first n odd positive integers
1. Jul 17, 2011
### pc2-brazil
1. The problem statement, all variables and given/known data
Give a big-O estimate of the product of the first n odd
positive integers.
2. Relevant equations
Big-O notation:
f(x) is O(g(x)) if there are constants C and k such that
|f(x)| ≤ C|g(x)| whenever x > k.
3. The attempt at a solution
The product of the first n odd integers can be given by:
$$P(n)=1\times 3\times 5\times 7\times...\times (2n-1)$$
For n > 0, no element in the above sequence will be greater than (2n-1). Thus:
$$1\times 3\times 5\times 7\times...\times (2n-1)\leq (2n-1)\times (2n-1)...\times (2n-1)=(2n-1)^n$$
So:
P(n) ≤ (2n-1)n whenever n > 0
I could stop here and say that
P(n) is O((2n-1)n)
But to simplify I think I could consider that:
P(n) ≤ (2n-1)n ≤ (2n)n
Thus,
P(n) is O((2n)n)
Is this reasoning correct?
Last edited: Jul 17, 2011
2. Jul 17, 2011
### tiny-tim
hi pc2-brazil!
it's correct, but it's not very accurate, is it?
do you know a big-O estimate for n! ?
3. Jul 17, 2011
### pc2-brazil
A big-O estimate for n! would be O(nn).
I could say that, for n > 0,
$$1\times 3\times 5\times 7...\times (2n-1)\leq 1\times 2\times 3\times 4\times...\times (2n-1)=(2n-1)!\leq (2n)!=2^n n!$$
Thus, P(n) is O(2nn!). Since n! is O(nn), this estimate seems more accurate than the previous one (O(2nnn)).
4. Jul 17, 2011
### tiny-tim
Last edited by a moderator: Apr 26, 2017
5. Jul 17, 2011
### Ray Vickson | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8989180604863166,
"lm_q2_score": 0.9136765269395709,
"openwebmath_perplexity": 2055.7448349221136,
"openwebmath_score": 0.8235124945640564,
"tags": null,
"url": "https://www.physicsforums.com/threads/give-a-big-o-estimate-of-the-product-of-the-first-n-odd-positive-integers.514750/"
} |
### tiny-tim
Last edited by a moderator: Apr 26, 2017
5. Jul 17, 2011
### Ray Vickson
If En = product of the even numbers from 2 to 2n - 2, your product is (2n-1)!/En, and En = 2^(n-1) * (n-1)! Now apply Stirling's formula to both factorials. Note: if you want a true upper bound, rather than just an *estimate* you can use the fact that if St(n) is defined as
sqrt(2pi)*n^(n + 1/2)*exp(-n), then we have St(n) <n! < St(n)*exp(1/(12n)), even if n is not large. You can use the upper bound on (2n-1)! in the numerator and the lower bound on (n-1)! in the denominator.
RGV
sqrt(2pi)*n | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8989180604863166,
"lm_q2_score": 0.9136765269395709,
"openwebmath_perplexity": 2055.7448349221136,
"openwebmath_score": 0.8235124945640564,
"tags": null,
"url": "https://www.physicsforums.com/threads/give-a-big-o-estimate-of-the-product-of-the-first-n-odd-positive-integers.514750/"
} |
rational numbers symbol
# rational numbers symbol | {
"domain": "calspasblog.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9935117296348998,
"lm_q1q2_score": 0.8987809095426977,
"lm_q2_score": 0.9046505267461572,
"openwebmath_perplexity": 321.2912625395907,
"openwebmath_score": 0.8266318440437317,
"tags": null,
"url": "https://calspasblog.com/z6oue/viewtopic.php?page=67a3e4-rational-numbers-symbol"
} |
The real line consists of the union of the rational and irrational numbers. We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. Formally, rational numbers are the set of all real numbers that can be written as a ratio of integers with nonzero denominator. The OP asked in his title only, if he can use the symbol for rational numbers. Hence, we can say that ‘0’ is also a rational number, as we can represent it in many forms such as 0/1, 0/2, 0/3, etc. In other words, most numbers are rational numbers. Answer: yes. Some examples of irrational numbers are $$\sqrt{2},\pi,\sqrt[3]{5},$$ and for example $$\pi=3,1415926535\ldots$$ comes from the relationship between the length of a circle and its diameter. Numbers that are not rational are called irrational numbers. The venn diagram below shows examples of all the different types of rational, irrational numbers including integers, whole numbers, repeating decimals and more. When we put together the rational numbers and the irrational numbers, we get the set of real numbers. Definition: Can be expressed as the quotient of two integers (ie a fraction) with a denominator that is not zero.. By using this website, you agree to our Cookie Policy. A rational number is a number that can be expressed as a fraction p/q where p and q are integers and q!=0. Rational numbers are indicated by the symbol . For instance, the decimal version of the therefore symbol (∴) would be ∴ The hexadecimal version of the therefore symbol (∴) would be ∴ Note that the hexadecimal numbers include x as part of the code. In Maths, rational numbers are represented in p/q form where q is not equal to zero. RapidTables. 1 $\begingroup$ I think your answer is fine. Set symbols of set theory and probability with name and definition: set, subset, union, intersection, element, cardinality, empty set, natural/real/complex number set. $\endgroup$ – Dietrich Burde Aug 15 '19 | {
"domain": "calspasblog.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9935117296348998,
"lm_q1q2_score": 0.8987809095426977,
"lm_q2_score": 0.9046505267461572,
"openwebmath_perplexity": 321.2912625395907,
"openwebmath_score": 0.8266318440437317,
"tags": null,
"url": "https://calspasblog.com/z6oue/viewtopic.php?page=67a3e4-rational-numbers-symbol"
} |
cardinality, empty set, natural/real/complex number set. $\endgroup$ – Dietrich Burde Aug 15 '19 at 18:42. Rational Numbers All positive and negative fractions, including integers and so-called improper fractions. Irrational numbers are a separate category of their own. Many people are surprised to know that a repeating decimal is a rational number. I just disagree that the notations should not be used for fields. Note that the set of irrational numbers is the complementary of the set of rational numbers. Expressed as an equation, a rational number is a number. where a and b are both integers. This equation shows that all integers, finite decimals, and repeating decimals are rational numbers. The denominator in a rational number cannot be zero. ... rational numbers set = … The ancient greek mathematician Pythagoras believed that all numbers were rational, but one of his students Hippasus proved (using geometry, it is thought) that you could not write the square root of 2 as a fraction, and so it was irrational. a/b, b≠0. It is also a type of real number. Top of Page The letters R, Q, N, and Z refers to a set of numbers such that: R = real numbers includes all real number [-inf, inf] Q= rational numbers ( numbers written as ratio) The Unicode numeric entity codes can be expressed as either decimal numbers or hexadecimal numbers. A rational number p/q is said to have numerator p and denominator q. Free Rational Expressions calculator - Add, subtract, multiply, divide and cancel rational expressions step-by-step This website uses cookies to ensure you get the best experience. Any fraction with non-zero denominators is a rational number. | {
"domain": "calspasblog.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9935117296348998,
"lm_q1q2_score": 0.8987809095426977,
"lm_q2_score": 0.9046505267461572,
"openwebmath_perplexity": 321.2912625395907,
"openwebmath_score": 0.8266318440437317,
"tags": null,
"url": "https://calspasblog.com/z6oue/viewtopic.php?page=67a3e4-rational-numbers-symbol"
} |
# Conditional probability exercise - apples and oranges
We have two crates, crate 1 and crate 2. Crate 1 has 2 oranges and 4 apples, and crate 2 has 1 orange and 1 apple. We take 1 fruit from crate 1 and put it in crate 2, and then we take a fruit from crate 2.
The first point of this exercise asks me to calculate the probability that the fruit taken from crate 2 is an orange. I did this by calculating the probability that the fruit we took from crate 1 was an orange(which is $\frac{2}{6}$) and then saying that I have 3 fruits in crate 2, $1+\frac{2}{6}$ oranges and the rest apples, which lead me to a $44.44\%$ probability that the fruit we take from crate 2 was an orange. The probability I got seems reasonable, but I don't know for sure if what I did was correct.
Anyway, point 2 of this problem is a little bit harder and I'm stuck. It tells me to calculate the probability that the fruit we took from crate 1 was an orange, if we know that the fruit we took out from crate 2 was also an orange. So if I consider A: Fruit taken from crate 1 was an orange, and B: Fruit taken from crate 2 was an orange, I think I have to calculate $\:P\left(A|B\right)$ I think, which means "Probability that A happens if we know B happened", but I'm not so sure about this.
Could anyone give me a hint on how to go about solving this problem? | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9877587261496031,
"lm_q1q2_score": 0.8987685918349758,
"lm_q2_score": 0.909907012756525,
"openwebmath_perplexity": 136.17207693072405,
"openwebmath_score": 0.6868855953216553,
"tags": null,
"url": "https://math.stackexchange.com/questions/2119158/conditional-probability-exercise-apples-and-oranges/2119210"
} |
Could anyone give me a hint on how to go about solving this problem?
• Your answer to the first question is correct. For the second question, $$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$$ where $A$ is the event that an orange was selected from the first crate and $B$ is the event that an orange was selected from the second crate. – N. F. Taussig Jan 29 '17 at 12:18
• But what is $P\left(A\cap B\right)$ ? I mean, I know it is the probability of the joint events, but how do I determine that? – MikhaelM Jan 29 '17 at 12:32
• It is the probability that you put an orange from the first crate into the second, and having done that then took an orange from the second . $$\mathsf P(A\cap B)= \mathsf P(A)~\mathsf P(B\mid A)$$ – Graham Kemp Jan 29 '17 at 12:34
• Doing that I get $\:P\left(A|B\right)=\frac{P\left(A\cap B\right)}{P\left(B\right)}=\frac{P\left(A\right)\cdot P\left(B|A\right)}{P\left(B\right)}=\frac{\frac{2}{6}\cdot \frac{2}{3}}{\frac{4}{9}}=\frac{1}{2}$ so $50\%$. Is this correct or am I making a mistake somewhere? – MikhaelM Jan 29 '17 at 12:45
• The answer you obtained in your comment is correct. – N. F. Taussig Jan 29 '17 at 13:15
Let $A,B$ be the events of removing an orange from the first and second crates, respectively.
You have calculated $\mathsf P(B) = 4/9$ correctly. Another way to look at it is through the law of total probability. \begin{align}\mathsf P(B) ~&=~\mathsf P(A)~\mathsf P(B\mid A)+\mathsf P(A^\complement)~\mathsf P(B\mid A^\complement) \\ &=~ \tfrac 26\cdot\tfrac 23+\tfrac 46\cdot\tfrac 13 & =&~ \frac{\tfrac 2 6+1}3 \\ &=~ \tfrac 49 \end{align} | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9877587261496031,
"lm_q1q2_score": 0.8987685918349758,
"lm_q2_score": 0.909907012756525,
"openwebmath_perplexity": 136.17207693072405,
"openwebmath_score": 0.6868855953216553,
"tags": null,
"url": "https://math.stackexchange.com/questions/2119158/conditional-probability-exercise-apples-and-oranges/2119210"
} |
Where $\mathsf P(A)$ is the probability of taking an orange from cart 1, $\mathsf P(A^\complement)$ is that of taking an apple from cart 1, $\mathsf P(B\mid A)$ is the probability of taking a orange from cart 2 when given that you have added an orange to that cart, and $\mathsf P(B\mid A^\complement)$ is the probability of taking a orange from cart 2 when given that you have added an apple to that cart.
Now you just need to calculate $\mathsf P(A\mid B)$ the probability of having taken an orange from cart 1 when given that you took an orange from cart 2.
Use Bayes' Rule.
Yes for second part solve P(A|B).
Probability of A happens if we know B happened. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9877587261496031,
"lm_q1q2_score": 0.8987685918349758,
"lm_q2_score": 0.909907012756525,
"openwebmath_perplexity": 136.17207693072405,
"openwebmath_score": 0.6868855953216553,
"tags": null,
"url": "https://math.stackexchange.com/questions/2119158/conditional-probability-exercise-apples-and-oranges/2119210"
} |
# general solution for a recurrence relation
I have the following recurrence relation: $$x_1=1, x_2=a, x_{n+2}=ax_{n+1}-x_n\hspace{1cm}(*)$$
If we assume that $x_n=r^n$ is a solution for the relation $x_{n+2}=ax_{n+1}-x_n$, then I can deduce that $r=\frac{a+\sqrt{a^2-4}}{2}$ or $r=\frac{a-\sqrt{a^2-4}}{2}$.
By using the initial values $x_1=1, x_2=a$, I found that $$x_n=\frac{1}{\sqrt{a^2-4}}\left(\frac{a+\sqrt{a^2-4}}{2}\right)^n-\frac{1}{\sqrt{a^2-4}}\left(\frac{a-\sqrt{a^2-4}}{2}\right)^n$$ is a solution for the recurrence relation (*).
Question: How do we know whether this is the only solution for the recurrence relation $(*)$? Notice that when I found the particular solution above I assumed that the solution was a linear combination of geometric series. But I do not know if all the solutions will have this form.
• Use method of difference on the recursive definition $x_{n+2}-x_{n+1}=(a-2)x_{n+1}+x_{n+1}-x_{n}$ and add these equations for all values of n. – Pi_die_die Jul 8 '18 at 20:57
• In other words, prove that it satisfies condition $(*)$. – steven gregory Jul 8 '18 at 20:59
• Yes you would get the sum which would be the type of sum you would expect from a gp – Pi_die_die Jul 8 '18 at 21:01
You have specific initial values $x_0$ and $x_1$. The recurrence relation fully determines all other values. Thus, there's exactly one solution. If you've found it, that's it, there can't be any others, no matter what approach you took in order to find it.
If you want to show that all solutions of $x_{n+2}=ax_{n+1}-x_n$ are combinations of the geometric series you found, you can argue as follows: | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9863631675246404,
"lm_q1q2_score": 0.8987541037619194,
"lm_q2_score": 0.9111797088058519,
"openwebmath_perplexity": 176.84608042968154,
"openwebmath_score": 0.8752542734146118,
"tags": null,
"url": "https://math.stackexchange.com/questions/2844980/general-solution-for-a-recurrence-relation"
} |
This is a linear second-order recurrence. Its solutions form a two-dimensional vector space. (A vector space because of the linearity of the recurrence, and a two-dimensional one because the solution space is spanned by the two solutions for the initial conditions $x_1=1$, $x_2=0$ and $x_1=0$, $x_2=1$.) You've found two linearly independent solutions; hence they span the entire solution space. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9863631675246404,
"lm_q1q2_score": 0.8987541037619194,
"lm_q2_score": 0.9111797088058519,
"openwebmath_perplexity": 176.84608042968154,
"openwebmath_score": 0.8752542734146118,
"tags": null,
"url": "https://math.stackexchange.com/questions/2844980/general-solution-for-a-recurrence-relation"
} |
• Hi @joriki I would appreciate some clarifications before I accept your answer. I understand that the set of solutions is a subspace of the vector space of sequences $\{f:\mathbb{N}\to\mathbb{C}: f\mbox{ is a function}\}$. I also understand that I found two solutions that are linearly independent. I do not know why the dimension of the solution space is 2 and not greater than 2. Also, what do you mean by $a_0$ and $a_1$? – Chilote Jul 8 '18 at 21:23
• @Chilote: I'm sorry, I meant $x_1$ and $x_2$; I've fixed that. The solution for given initial conditions $x_1=\hat x_1$ and $x_2=\hat x_2$ is $\hat x_1s_1+\hat x_2 s_2$, where $s_1$ is the solution for $x_1=1$ and $x_2=0$ and $s_2$ is the solution for $x_1=0$ and $x_2=1$; so $s_1$ and $s_2$ span the solution space; hence it's at most two-dimensional (in fact exactly two-dimensional, since $s_1$ and $s_2$ are linearly independent). – joriki Jul 8 '18 at 21:31
• Got it, the solution space of $x_{n+2}=ax_{n+1}-x_n$ is $S=\{(x_n)_n:x_{n+2}=ax_{n+1}-x_n\}$ which is a vector space. Any vector in this space is of the form $x=(x_1,x_2,ax_2-x_1,a(ax_2-x_1)-x_2,\dots)=x_1 s_1+x_2 s_2$ where $s_1$ and $s_2$ are two particular solutions that are linearly independent. – Chilote Jul 8 '18 at 21:49
• You've found two linearly independent solutions Just to nitpick, that's iff $a \ne \pm 2$. – dxiv Jul 8 '18 at 23:29
• @Chilote: This was actually overkill, since you only asked about the uniqueness of one particular solution, not about all solutions being combinations of the geometric series. I edited the answer accordingly. – joriki Jul 9 '18 at 5:22
A bit of culture. Any two consecutive numbers in your sequence, call them $x_n$ and $x_{n+1},$ satisfy $$x_n^2 - a \, x_n \, x_{n+1} + x_{n+1}^2 = 1$$
Try consecutive values in $$1, \; \; a, \; \; a^2 - 1, \; \; a^3 - 2a, \; \ldots$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9863631675246404,
"lm_q1q2_score": 0.8987541037619194,
"lm_q2_score": 0.9111797088058519,
"openwebmath_perplexity": 176.84608042968154,
"openwebmath_score": 0.8752542734146118,
"tags": null,
"url": "https://math.stackexchange.com/questions/2844980/general-solution-for-a-recurrence-relation"
} |
Try consecutive values in $$1, \; \; a, \; \; a^2 - 1, \; \; a^3 - 2a, \; \ldots$$
This comes from the matrix $$\left( \begin{array}{cc} 0 & 1 \\ -1 & a \end{array} \right)$$ which has determinant $1$ and trace $a.$ It also gives an automorphism of the quadratic form $x^2 - a xy+y^2.$ An automorphism matrix $P$ for a quadratic form that has Hessian matrix $H$ satisfies $P^T HP = H$
In this case $$\left( \begin{array}{cc} 0 & -1 \\ 1 & a \end{array} \right) \left( \begin{array}{cc} 2 & -a \\ -a & 2 \end{array} \right) \left( \begin{array}{cc} 0 & 1 \\ -1 & a \end{array} \right) = \left( \begin{array}{cc} 2 & -a \\ -a & 2 \end{array} \right)$$
The explicit relation with the sequence is
$$\left( \begin{array}{cc} 0 & 1 \\ -1 & a \end{array} \right) \left( \begin{array}{c} x_n \\ x_{n+1} \end{array} \right) = \left( \begin{array}{c} x_{n+1} \\ x_{n+2} \end{array} \right)$$
• How can I get the first equality? – Chilote Jul 8 '18 at 21:52
• @Chilote the automorphism group for an (indefinite) binary quadratic form, along with the Pell type equation $\tau^2 - \Delta \sigma^2 = 4,$ is discussed in many number theory and quadratic forms books. – Will Jagy Jul 8 '18 at 23:24
• I think the quadratic form corresponding to the matrix you wrote above is $ay^2$ – Chilote Jul 8 '18 at 23:59
• @Chilote I put in a little about the automorphism. The matrix that gives the quadratic form is the symmetric one; it is just the Hessian matrix of second partial derivatives – Will Jagy Jul 9 '18 at 0:23 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9863631675246404,
"lm_q1q2_score": 0.8987541037619194,
"lm_q2_score": 0.9111797088058519,
"openwebmath_perplexity": 176.84608042968154,
"openwebmath_score": 0.8752542734146118,
"tags": null,
"url": "https://math.stackexchange.com/questions/2844980/general-solution-for-a-recurrence-relation"
} |
Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share ⦠In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. If P A Ais nonsingular then the eigenvectors of P 1 U Acorresponding to are of the form [0 T;vT] where v is any eigenvector of P 1 S Cthat corresponds to its unit eigenvalue. T is diagonal iff A is symmetric. Every square real matrix A is orthogonally similar to an upper block triangular matrix T with A=Q T TQ where each block of T is either a 1#1 matrix or a 2#2 matrix having complex conjugate eigenvalues. Moreover, the eigenvectors of P 1 U Acorresponding to are of the form [uT;((P S+ C) 1Bu) T] . Block lower triangular matrices and block upper triangular matrices are popular preconditioners for $2\times 2$ block matrices. Moreover, the eigenvectors of P 1 Based on the lemma, we can derive the following main results about the SBTS iteration method. Yes. First of all: what is the determinant of a triangular matrix? However, a 2 by 2 symmetric matrix cannot have imaginary eigenvalues, so R must be diagonal. The second consequence of Schurâs theorem says that every matrix is similar to a block-diagonal matrix where each block is upper triangular and has a constant diagonal. The determinant of a block-diagonal matrix is the product of the determinants of the blocks, so, by considering the definition of the characteristic polynomial, it should be clear that the eigenvalues of a block-diagonal matrix are the eigenvalues of the blocks. Assume that α is a positive constant and S = W â 1 T. These eigenvectors form an orthonormal set. Theorem 3.2. Then the eigenvalues of the matrix S = W â 1 T are all real, and S is similar to a diagonal matrix. 2 AQ = QÎ A(Qe i)=(Qe i)λ i Qe i is an eigenvector, and λ i is eigenvalue. TRIANGULAR PRECONDITIONED BLOCK | {
"domain": "afd-hamburg-nord.de",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9949396688683126,
"lm_q1q2_score": 0.8987232330807353,
"lm_q2_score": 0.9032942008463507,
"openwebmath_perplexity": 430.25903240845196,
"openwebmath_score": 0.786680281162262,
"tags": null,
"url": "https://afd-hamburg-nord.de/forum/archive.php?53e5c2=block-triangular-matrix-eigenvalues"
} |
A(Qe i)=(Qe i)λ i Qe i is an eigenvector, and λ i is eigenvalue. TRIANGULAR PRECONDITIONED BLOCK MATRICES 3 P 1 A Athat corresponds to its unit eigenvalue. Theorem 6. This is an important step in a possible proof of Jordan canonical form. Developing along the first column you get $a_{11} \det(A_{11}'),$ where $A_{11}'$ is the minor you get by crossing out the first row and column of [math]A. This method can be impractical, however, due to the contamination of smaller eigenvalues by This decouples the problem of computing the eigenvalues of Ainto the (solved) problem of computing 1, and then computing the remaining eigenvalues by focusing on the lower right (n 1) (n 1) submatrix. Let W, T â R n × n be symmetric positive definite and symmetric, respectively. Hence R is symmetric block diagonal with blocks that either are 1 by 1 or are symmetric and 2 by 2 with imaginary eigenvalues. upper-triangular, then the eigenvalues of Aare equal to the union of the eigenvalues of the diagonal blocks. If each diagonal block is 1 1, then it follows that the eigenvalues of any upper-triangular matrix are the diagonal elements. 1 is a matrix with block upper-triangular structure. Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or partition it, into a collection of smaller matrices. In this note we show that a block lower triangular preconditioner gives the same spectrum as a block upper triangular preconditioner and that the eigenvectors of the two preconditioned matrices are related. | {
"domain": "afd-hamburg-nord.de",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9949396688683126,
"lm_q1q2_score": 0.8987232330807353,
"lm_q2_score": 0.9032942008463507,
"openwebmath_perplexity": 430.25903240845196,
"openwebmath_score": 0.786680281162262,
"tags": null,
"url": "https://afd-hamburg-nord.de/forum/archive.php?53e5c2=block-triangular-matrix-eigenvalues"
} |
Summer Film 2019, Denon Avr-x2600h Bundle, Heavy Duty Workbench Brackets, Naruto Vs Pain Episodes, Saturday Kitchen Vote, Rent Townhouse By Owner, Book Template Word, | {
"domain": "afd-hamburg-nord.de",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9949396688683126,
"lm_q1q2_score": 0.8987232330807353,
"lm_q2_score": 0.9032942008463507,
"openwebmath_perplexity": 430.25903240845196,
"openwebmath_score": 0.786680281162262,
"tags": null,
"url": "https://afd-hamburg-nord.de/forum/archive.php?53e5c2=block-triangular-matrix-eigenvalues"
} |
# A Seemingly Impossible Problem
Given that $w, x, y, z$ take on values $0$ and $1$ with equal probability, what is the probability that $w+x+y+z$ is odd?
Which of the following arguments is correct?
Furthermore, can you generalize this result?
Argument 1:
If all 4 numbers are even, the sum is even.
If 3 numbers are even, the sum is odd.
If 2 numbers are even, the sum is even.
If 1 number is even, the sum is odd.
If 0 numbers are even, the sum is even.
In 2 of the 5 cases, the sum is odd. Hence, $w + x +y + z$ is odd with probability $\frac{ 2}{5}$.
Argument 2:
$w + x + y + z$ is either odd or even.
In 1 of the 2 cases, the sum is odd. Hence, $w + x +y + z$ is odd with probability $\frac{ 1}{2}$.
Argument 3:
By listing out all the $2^4$ cases, we find that there are ${ 4 \choose 0} + { 4 \choose 2} + { 4 \choose 4 } = 8$ cases with an even sum, and ${ 4 \choose 1} + {4 \choose 3} = 8$ cases with an odd sum.
Hence, $w + x +y + z$ is odd with probability $\frac{ 1}{2}$.
Context: In a recent problem, Argument 1 was given, and many people agreed with it.
Note by Calvin Lin
5 years, 5 months ago
This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.
When posting on Brilliant: | {
"domain": "brilliant.org",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9759464485047916,
"lm_q1q2_score": 0.8986414150663654,
"lm_q2_score": 0.9207896769778074,
"openwebmath_perplexity": 1416.1759739498716,
"openwebmath_score": 0.968029260635376,
"tags": null,
"url": "https://brilliant.org/discussions/thread/a-seemingly-impossible-problem/"
} |
When posting on Brilliant:
• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
• Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.
MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2
paragraph 1
paragraph 2
[example link](https://brilliant.org)example link
> This is a quote
This is a quote
# I indented these lines
# 4 spaces, and now they show
# up as a code block.
print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.
print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$
Sort by:
The problem with the first argument is that the probabilities of each of the cases is different. In reality, the first case has one way, the second has 4 ways, the third has 6 ways, the fourth has 4 ways, and the last one has only 1 way. Thus the actual probability is $\dfrac{4+4}{1+4+6+4+1}=\dfrac{1}{2}$.
- 5 years, 5 months ago
I disagree with Argument 1:
In case 1 there is only 1 outcome to make an even sum.
In case 2 there are 4 outcomes that make an odd sum | {
"domain": "brilliant.org",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9759464485047916,
"lm_q1q2_score": 0.8986414150663654,
"lm_q2_score": 0.9207896769778074,
"openwebmath_perplexity": 1416.1759739498716,
"openwebmath_score": 0.968029260635376,
"tags": null,
"url": "https://brilliant.org/discussions/thread/a-seemingly-impossible-problem/"
} |
In case 2 there are 4 outcomes that make an odd sum
In case 3 there are 6 outcomes that make an even sum
In case 4 there are 4 outcomes that make an odd sum
In case 5 there is only 1 outcome to make an even sum.
Thus out of the 16 equally likely outcomes 8 will yield an odd sum and 8 will yield an even sum.
Therefore the probability that w + x + y + z is odd is 1/2.
- 5 years, 5 months ago
Reminds me of Bertrand paradox, where three ways of choosing a seemingly the same uniformly random chord on a circle give different probability distributions (which means they are different).
- 5 years, 5 months ago
I am not very sure if these two are similar indeed. The problem stated above seems to have objective solution and other two are simply counting errors.
- 5 years, 3 months ago
They are not similar, yes. It's just the fact that there are three solutions arriving at different answers immediately reminded me to that.
- 5 years, 3 months ago
Well, this is how i solve this problem.
Assume that we choose w, x, y first.
w + x + y = A.
Doesn't matter whether A is odd or even. z will be the one that decide whether the sum of w, x, y, z (assume w +x+y+z= B) is odd or even
=> With that the possibility for B to be odd = 50%
=> Argument 1: Wrong
Argument 2: Correct (but lack at explaining)
Argument 3: Correct
- 5 years, 5 months ago
While argument 2 produces the correct numerical answer, the logic presented is incorrect.
See @SAMARTH M.O. Comment.
Staff - 5 years, 5 months ago
Argument 1 is wrong because it assumes they all happen with equal probability, which they don't.
- 5 years, 5 months ago
Argument 3 is arguably the right one. Arguments 1 and 2 assume that their mentioned respective events are equally likely.
- 5 years, 5 months ago
Was this problem perhaps my problem All That Glitters Is Gold?
- 5 years, 5 months ago
No. I like your problem, which highlights a common mistake made by those who are first introduced to probability. | {
"domain": "brilliant.org",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9759464485047916,
"lm_q1q2_score": 0.8986414150663654,
"lm_q2_score": 0.9207896769778074,
"openwebmath_perplexity": 1416.1759739498716,
"openwebmath_score": 0.968029260635376,
"tags": null,
"url": "https://brilliant.org/discussions/thread/a-seemingly-impossible-problem/"
} |
The problem that I was referencing had a similar title to my note. It has since been deleted.
Staff - 5 years, 5 months ago
Dude Calvin, I just found this. You didn't even tag me in it! But thanks, it's still awesome! :D
- 5 years, 5 months ago
The last argument is correct. In general for n numbers the probability is $\frac {{2}^{n-1}}{{2}^n} = \frac {1}{2}$
- 5 years, 5 months ago
argument 3 feels more logical
- 5 years, 5 months ago
.5
- 5 years, 5 months ago
Argument 3 is perfect.
- 5 years, 5 months ago
1\2
- 5 years, 5 months ago
2
- 5 years, 5 months ago
Obviously universal space= 2^4 Sample space= 4C0+ 4C2+ 4C4= 8 So probability= 8/16 =1/2
- 5 years, 5 months ago
let a(n) = P(Bin(n,0.5) is even), then a(n+1) = 0.5a(n) + (1-0.5)(1-a(n)) = 0.5
- 5 years, 5 months ago
We have 2^{4} case. Odd sum is 2.\frac{4!}{3!}=2^{3} case. There for sum is odd with probability \frac{1}{2}
- 5 years, 4 months ago
An even easier argument which is easily adapter to be generalisation for the number of: picking the first 3 numbers randomly, you'll find either an even or an uneven number. The last term will change it to either even or uneven, with an equal chance of either.
This will generalise to any number of variables easily, and also adapt to the odds of the result being k (mod n) if your variables can have values 1, 2, ... , n.
One problem it can't help with is: if the variables are binary, what are the odds of the result being a multiple of k (for a k greater than 2).
- 5 years, 4 months ago
1
- 5 years, 4 months ago
The first arguement is wrong as each case doesnot have equal probability .. The second solution (though gives right answer ) is an insufficient one. ARGUEMENT 3is descent
- 4 years, 7 months ago
The second solution doesn't give a correct argument. The point isn't to "get to the correct answer". The point is to "substantiate your argument".
Staff - 4 years, 7 months ago | {
"domain": "brilliant.org",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9759464485047916,
"lm_q1q2_score": 0.8986414150663654,
"lm_q2_score": 0.9207896769778074,
"openwebmath_perplexity": 1416.1759739498716,
"openwebmath_score": 0.968029260635376,
"tags": null,
"url": "https://brilliant.org/discussions/thread/a-seemingly-impossible-problem/"
} |
Staff - 4 years, 7 months ago
Total number of cases = 222*2 Even cases are If number of ONEs is 2 or 4, So Favourable number of cases : 4P2 + 4P4 = 6 + 1 = 7 So Probability for odd is (16 - 7)/16 = 9/16
- 4 years, 6 months ago
The answer is 1/2.
When dealing with even cases, you forgot that 0 is an even number.
Staff - 4 years, 6 months ago
Argument 1 is incorrect because each case is not equally likely. Argument 2 gives the correct answer but for the wrong reason. Consider the probability of drawing the Ace of spades from a deck of cards. The card drawn is either the ace or it is not. One out of two cases is favourable hence 1/2. (?) Argument 3 would be correct if it was asserted that each case was equally probable. Which of course it is. J
- 4 years, 6 months ago
Great :)
Staff - 4 years, 6 months ago
The issue with the first argument is that of frequency distribution
- 5 years, 5 months ago
now since every single variable counts and is equally important we should be going with the the third one .. ARGUMENT 3 is correct .... nearly similar way to solve is .. there are two possibilities:-
1.) - three variables are 1
2.)- only one variable is 1 .. both ways we get odd
hence- 4C3[ any three numbers] (1/2)^4[the probability of getting 1 is 1/2 and zero is also the same AND we need to get 1 in three cases AS WELL AS 0 in the other case] => 4C3 * (1/2)^4 +4C1[only one variable](1/2)^4 AND HENCE
4C3 * (1/2)^4 + 4C1 * (1/2)^4 = 1/2 ..
- 5 years, 5 months ago
1
- 5 years, 5 months ago
10
- 5 years, 5 months ago | {
"domain": "brilliant.org",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9759464485047916,
"lm_q1q2_score": 0.8986414150663654,
"lm_q2_score": 0.9207896769778074,
"openwebmath_perplexity": 1416.1759739498716,
"openwebmath_score": 0.968029260635376,
"tags": null,
"url": "https://brilliant.org/discussions/thread/a-seemingly-impossible-problem/"
} |
What is the remainder of $(14^{2010}+1) \div 6$?
What is the remainder of $(14^{2010}+1) \div 6$?
Someone showed me a way to do this by finding a pattern, i.e.:
$14^1\div6$ has remainder 2
$14^2\div6$ has remainder 4
$14^3\div6$ has remainder 2
$14^4\div6$ has remainder 4
And it seems that when the power is odd, the answer is 2, and when it's even, the answer is 4.
2010 is even, so the remainder is 4, but we have that +1, so the final remainder is 5. Which is correct.
But this method doesn't seem very concrete to me, and I have a feeling the pattern may not be easy to find (or exist?) for every question. What theorem or algorithm can I use to solve this instead?
-
$14\equiv 2\pmod 6$, so $14^{2010}+1\equiv 2^{2010}+1\pmod 6$. Now $2\cdot 2^2=2^3=8\equiv 2\pmod 6$, so $2\cdot (2^2)^k\equiv 2\pmod 6$ for any non-negative integer $k$. This shows that the pattern that you observed is real: $2^{2k+1} \equiv 2\pmod 6$ for any non-negative integer $k$. In particular, $$2^{2010}+1\equiv 2\cdot 2^{2009}+1 \equiv 2\cdot 2+1 \equiv 5\pmod 6\;.$$
The same basic idea can be used in similar problems, though the cycle of the pattern may not be nearly so short. To go much deeper than this kind of analysis, you want to look into the Chinese Remainder Theorem.
-
$14=0\pmod{2}$ and $14=2\pmod{3}$. Thus, because $2^2=1\pmod{3}$, for any $k>0$, we have $$14^k=0\pmod{2}$$ and $$14^k=\left\{\begin{array}{}1\pmod{3}&\text{when }k=0\pmod{2}\\2\pmod{3}&\text{when }k=1\pmod{2}\end{array}\right.$$ Therefore, for any $k>0$, we have by the Chinese Remainder Theorem, $$14^k=\left\{\begin{array}{}4\pmod{6}&\text{when }k=0\pmod{2}\\2\pmod{6}&\text{when }k=1\pmod{2}\end{array}\right.$$ So, as you surmised, $14^{2010}+1=5\pmod{6}$.
-
To solve this problem, in general, I'd use use two facts: | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.980875959894864,
"lm_q1q2_score": 0.8985899055847951,
"lm_q2_score": 0.9161096227509861,
"openwebmath_perplexity": 175.36066282774567,
"openwebmath_score": 0.782090425491333,
"tags": null,
"url": "http://math.stackexchange.com/questions/79446/what-is-the-remainder-of-1420101-div-6"
} |
-
To solve this problem, in general, I'd use use two facts:
1. If $a$ and $n$ are relatively prime, $a^{\phi(n)} \bmod n = 1$. Here $\phi$ is the Euler phi function; when $n$ is prime, $\phi(n) = n-1$. This lets you reduce exponents to ones with power less than $\phi(n)$; for instance $$2^{2010} \bmod 5 = (2^4)^{502} 2^2 \bmod 5 = 1^{502} 4 \bmod 5 = 4.$$ (Here $2^2$ happened to be trivial to compute by hand; if the exponent were larger I would calculate it using the technique of repeated squaring.)
2. Unfortunately 2 and 6 are not relatively prime; in this case I would factor $6$ into primes and use the Chinese remainder theorem: $$2^{2010} \bmod 3 = (2^2)^{1005} \bmod 3 = 1$$ $$2^{2010} \bmod 2 = 0$$ so applying the Chinese remainder theorem, $2^{2010}$ must be congruent to 4 mod 6.
-
We may write following equalities :
$14^{2010}=6\cdot k_1+4$
$14^{2010}+1=6\cdot k_2+r$
$6\cdot k_1+4+1=6\cdot k_2+r\Rightarrow 6k_1+5=6k_2+r$
The last equality is true only if $k_1=k_2$ and $r=5$
-
Please excuse my ignorance, but it looks to me like you assumed that $14^{2010} = 4 \mod 6$, and then use this to 'prove' that $14^{2010} + 1 = 5 \mod 6$. How do you justify the first equality given $k_1$ is an integer? – tom Nov 6 '11 at 9:01
@tom,read carefully text of the question... – pedja Nov 6 '11 at 10:03
HINT $\rm\quad (m,n) = 1,\ m\:|\:a,\ n\:|\:a+1\ \ \Rightarrow\ \ a^{2\:k}\ \equiv\ m\:(m^{-1}\ mod\ n)\ \pmod{mn}\$ by CRT.
Therefore for $\rm\:m,n,a\ =\ 2,3,14\:$ we infer $\rm\: 14^{\:2\:k} \equiv\: 2\ (2^{-1}\ mod\ 3)\equiv -2\pmod 6$
- | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.980875959894864,
"lm_q1q2_score": 0.8985899055847951,
"lm_q2_score": 0.9161096227509861,
"openwebmath_perplexity": 175.36066282774567,
"openwebmath_score": 0.782090425491333,
"tags": null,
"url": "http://math.stackexchange.com/questions/79446/what-is-the-remainder-of-1420101-div-6"
} |
# What's the probability of “at least” and “exactly” one event occurring?
If I know the probability of event $A$ occurring and I also know the probability of $B$ occurring, how can I calculate the probability of "at least one of them" occurring?
I was thinking that this is $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and }B)$.
Is this correct?
If it is, then how can I solve the following problem taken from DeGroot's Probability and Statistics:
If $50$ percent of families in a certain city subscribe to the morning newspaper, $65$ percent of the families subscribe to the afternoon newspaper, and $85$ percent of the families subscribe to at least one of the two newspapers, what proportion of the families subscribe to both newspapers?
In a more mathematical language, we are given $P(\text{morning})=.5$, $P(\text{afternoon})=.65$, $P(\text{morning or afternoon}) = .5 + .65 - P(\text{morning and afternoon}) = .85$, which implies that $P(\text{morning and afternoon}) = .3$, which should be the answer to the question.
Is my reasoning correct?
If it is correct, how can I calculate the following?
If the probability that student $A$ will fail a certain statistics examination is $0.5$, the probability that student $B$ will fail the examination is $0.2$, and the probability that both student $A$ and student $B$ will fail the examination is $0.1$, what is the probability that exactly one of the two students will fail the examination?
These problems and questions highlight the difference between "at least one of them" and "exactly one of them". Provided that "at least one of them" is equivalent to $P(A \text{ or } B)$, but how can I work out the probability of "exactly one of them"?
You are correct.
To expand a little: if $A$ and $B$ are any two events then
$$P(A\textrm{ or }B) = P(A) + P(B) - P(A\textrm{ and }B)$$
or, written in more set-theoretical language,
$$P(A\cup B) = P(A) + P(B) - P(A\cap B)$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9918120914294177,
"lm_q1q2_score": 0.8985716642327322,
"lm_q2_score": 0.905989826094673,
"openwebmath_perplexity": 273.2812882762553,
"openwebmath_score": 0.819298505783081,
"tags": null,
"url": "https://math.stackexchange.com/questions/72589/whats-the-probability-of-at-least-and-exactly-one-event-occurring"
} |
or, written in more set-theoretical language,
$$P(A\cup B) = P(A) + P(B) - P(A\cap B)$$
In the example you've given you have $A=$ "subscribes to a morning paper" and $B=$ "subscribes to an afternoon paper." You are given $P(A)$, $P(B)$ and $P(A\cup B)$ and you need to work out $P(A\cap B)$ which you can do by rearranging the formula above, to find that $P(A\cap B) = 0.3$, as you have already worked out.
• "Exactly one of A and B" means "Either A or B, but not both" which you can calculate as P(A or B) - P(A and B). – Chris Taylor Oct 14 '11 at 11:13
• Are you asking about the notation itself, or the method of displaying the notation? To write the notation we use $\LaTeX$ - you can find a tutorial by searching for "latex tutorial" in Google. Here's one, for example. If you want to learn the notation itself, the best way is learning by doing. You should read a mathematics text that's appropriate for your level, and make sure you understand all the notation used there. As you read more complex texts, you will become more and more familiar with the notation. – Chris Taylor Oct 14 '11 at 11:21
• So if I download LaTeX and paste your notation then it displays it in a more readable form? – upabove Oct 14 '11 at 11:24
For your second question, you know $\Pr(A)$, $\Pr(B)$, and $\Pr(A \text{ and } B)$, so you can work out $\Pr(A \text{ and not } B)$ and $\Pr(B \text{ and not } A)$ by taking the differences. Then add these two together.
Alternatively take $\Pr(A \text{ or } B) - \Pr(A \text{ and } B)$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9918120914294177,
"lm_q1q2_score": 0.8985716642327322,
"lm_q2_score": 0.905989826094673,
"openwebmath_perplexity": 273.2812882762553,
"openwebmath_score": 0.819298505783081,
"tags": null,
"url": "https://math.stackexchange.com/questions/72589/whats-the-probability-of-at-least-and-exactly-one-event-occurring"
} |
Alternatively take $\Pr(A \text{ or } B) - \Pr(A \text{ and } B)$.
• Pr(A and not B) + Pr(B and not A) is not the same as Pr(A) + Pr(B) - Pr(A and B) – Petr Peller Apr 4 '15 at 17:44
• Pr(A and not B) + Pr(B and not A) + Pr(A and B) is what you are looking for, but calculating Pr(A) + Pr(B) - Pr(A and B) seems to be much easier. – Petr Peller Apr 4 '15 at 18:06
• $\Pr(A \text{ and not } B)+\Pr(B \text{ and not } A)$ is the answer to "but how can I work out the probability of exactly one of them?" which is what I meant by "your second question" – Henry Apr 4 '15 at 18:08
For the additional problem: probability of exactly one equals probability of one or the other but not both, equals probability of union minus probability of intersection, equals $$P(A)+P(B)-2P(A\cap B)$$
probability of only one event occuring is as follows: if A and B are 2 events then probability of only A occuring can be given as P(A and B complement)= P(A) - P(A AND B ) | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9918120914294177,
"lm_q1q2_score": 0.8985716642327322,
"lm_q2_score": 0.905989826094673,
"openwebmath_perplexity": 273.2812882762553,
"openwebmath_score": 0.819298505783081,
"tags": null,
"url": "https://math.stackexchange.com/questions/72589/whats-the-probability-of-at-least-and-exactly-one-event-occurring"
} |
One property I am aware of is that $AA^H$ is Hermitian, i.e. An matrix can be multiplied on the right by an matrix, where is any positive integer. If you want to discuss contents of this page - this is the easiest way to do it. It only takes a minute to sign up. To learn more, see our tips on writing great answers. The difference of a square matrix and its conjugate transpose ( A − A H ) {\displaystyle \left(A-A^{\mathsf {H}}\right)} is skew-Hermitian (also called antihermitian). What special properties are possessed by $AA^H$, where $^H$ denotes the conjugate transpose? Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. In this representation, the conjugate of a quaternion corresponds to the transpose of the matrix. Wikidot.com Terms of Service - what you can, what you should not etc. Thanks for contributing an answer to Mathematics Stack Exchange! Matrix multiplication error in conjugate transpose. How to create a geometry generator symbol using PyQGIS, Does fire shield damage trigger if cloud rune is used. Yes. Check out how this page has evolved in the past. Learn more about multiplication error, error using *, incorrect dimensions After 20 years of AES, what are the retrospective changes that should have been made? The complex conjugate transpose of a matrix interchanges the row and column index for each element, reflecting the elements across the main diagonal. A matrix math implementation in python. To perform elementwise Note that A ∗ represents A adjoint, i.e. Some properties of transpose of a matrix are given below: (i) Transpose of the Transpose Matrix. A ComplexHermitianMatrix that is the product of this ComplexDenseMatrix with its conjugate transpose. We are about to look at an important theorem which will give us a relationship between a matrix that represents the linear transformation $T$ and a matrix that | {
"domain": "charlottegroutars.nl",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9886682478041813,
"lm_q1q2_score": 0.8983216640840999,
"lm_q2_score": 0.9086178969328287,
"openwebmath_perplexity": 714.0621717159065,
"openwebmath_score": 0.5960926413536072,
"tags": null,
"url": "https://charlottegroutars.nl/f960gm/9a14f6-matrix-multiplied-by-its-conjugate-transpose"
} |
us a relationship between a matrix that represents the linear transformation $T$ and a matrix that represents the adjoint of $T$, $T^*$. Hot Network Questions Can you make a CPU out of electronic components drawn by hand on paper? Here are the matrices: And here is what I am trying to calculate: The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2×2 real matrices, obeying matrix addition and multiplication: A square complex matrix whose transpose is equal to the matrix with every entry replaced by its complex conjugate (denoted here with an overline) is called a Hermitian matrix (equivalent to the matrix being equal to its conjugate transpose); that is, A is Hermitian if {\displaystyle \mathbf {A} ^ {\operatorname {T} }= {\overline {\mathbf {A} }}.} Definition of Spectral Radius / Eigenvalues of Product of a Matrix and its Complex Conjugate Transpose 1 Properties of the product of a complex matrix with its complex conjugate transpose Matrix Transpose. At whose expense is the stage of preparing a contract performed? Find out what you can do. The conjugate transpose of A is also called the adjoint matrix of A, the Hermitian conjugate of A (whence one usually writes A ∗ = A H). Are there any other special properties of $AA^H$? Change the name (also URL address, possibly the category) of the page. Solving a matrix equation involving transpose conjugates. Making statements based on opinion; back them up with references or personal experience. Two matrices can only be added or subtracted if they have the same size. Check that the number of columns in the first matrix matches the number of rows in the second matrix. A = [ 7 5 3 4 0 5 ] B = [ 1 1 1 − 1 3 2 ] {\displaystyle A={\begin{bmatrix}7&&5&&3\\4&&0&&5\end{bmatrix}}\qquad B={\begin{bmatrix}1&&1&&1\\-1&&3&&2\end{bmatrix}}} Here is an example of matrix addition 1. eigenvalues of sum of a matrix and its conjugate transpose, Solving a matrix equation involving transpose | {
"domain": "charlottegroutars.nl",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9886682478041813,
"lm_q1q2_score": 0.8983216640840999,
"lm_q2_score": 0.9086178969328287,
"openwebmath_perplexity": 714.0621717159065,
"openwebmath_score": 0.5960926413536072,
"tags": null,
"url": "https://charlottegroutars.nl/f960gm/9a14f6-matrix-multiplied-by-its-conjugate-transpose"
} |
of sum of a matrix and its conjugate transpose, Solving a matrix equation involving transpose conjugates. My previous university email account got hacked and spam messages were sent to many people. When 2 matrices of order (m×n) and (n×m) (m × n) and (n × m) are multiplied, then the order of the resultant matrix will be (m×m). site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. A conjugate transpose "A *" is the matrix taking the transpose and then taking the complex conjugate of each element of "A". Remarks. So if A is just a real matrix and A satisfies A t A = A A t, then A is a normal matrix, as the complex conjugate transpose of a real matrix is just the transpose of that matrix. Why do jet engine igniters require huge voltages? rev 2021.1.18.38333, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, Properties of the Product of a Square Matrix with its Conjugate Transpose. Some applications, for example the solution of a least squares problem using normal equations, require the product of a matrix with its own transpose… Incorrect dimensions for matrix multiplication. Is the determinant of a complex matrix the complex conjugate of the determinant of it's complex conjugate matrix? $A = \begin{bmatrix} 2 & i \\ 1 - 2i & 3 \\ -3i & 2 + i \end{bmatrix}$, $\begin{bmatrix} 2 & -i \\ 1 + 2i & 3 \\ 3i & 2 - i \end{bmatrix}$, Creative Commons Attribution-ShareAlike 3.0 License. Matrix transpose AT = 15 33 52 −21 A = 135−2 532 1 Example Transpose operation can be viewed as flipping entries about the diagonal. eigenvalues of sum of a matrix and its conjugate transpose. To print the transpose of the given matrix − Create an empty | {
"domain": "charlottegroutars.nl",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9886682478041813,
"lm_q1q2_score": 0.8983216640840999,
"lm_q2_score": 0.9086178969328287,
"openwebmath_perplexity": 714.0621717159065,
"openwebmath_score": 0.5960926413536072,
"tags": null,
"url": "https://charlottegroutars.nl/f960gm/9a14f6-matrix-multiplied-by-its-conjugate-transpose"
} |
a matrix and its conjugate transpose. To print the transpose of the given matrix − Create an empty matrix. Properties of transpose Why is “HADAT” the solution to the crossword clue "went after"? Milestone leveling for a party of players who drop in and out? The sum of a square matrix and its conjugate transpose (+) is Hermitian. Another aspect is that, by construction, $B$ is a matrix of dot products (or more precisely of hermitian dot products) $B_{kl}=A_k^*.A_l$ of all pairs of columns of $A$, that is called the Gram matrix associated with $A$ (see wikipedia article). In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric ) matrix is a square matrix whose transpose equals its negative. What do you call a 'usury' ('bad deal') agreement that doesn't involve a loan? 0. does paying down principal change monthly payments? Conjugate and transpose the first and third dimensions: ... Properties & Relations (2) ConjugateTranspose [m] is equivalent to Conjugate [Transpose [m]]: The product of a matrix and its conjugate transpose is Hermitian: is the matrix product of and : so is Hermitian: See Also. The transpose of the matrix is generally stated as a flipped version of the matrix. Definition The transpose of an m x n matrix A is the n x m matrix AT obtained by interchanging rows and columns of A, Definition A square matrix A is symmetric if AT = A. Notation. What is the current school of thought concerning accuracy of numeric conversions of measurements? Click here to toggle editing of individual sections of the page (if possible). Transpose of matrix M is represented by M T. There are numerous ways to transpose matrices.The transpose of matrices is basically done because they are used to represent linear transformation. A + B = [ 7 + 1 5 + 1 3 + 1 4 − 1 0 + 3 5 … General Wikidot.com documentation and help section. Append content without editing the whole page source. Then the conjugate transpose of $A$ is obtained by first taking | {
"domain": "charlottegroutars.nl",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9886682478041813,
"lm_q1q2_score": 0.8983216640840999,
"lm_q2_score": 0.9086178969328287,
"openwebmath_perplexity": 714.0621717159065,
"openwebmath_score": 0.5960926413536072,
"tags": null,
"url": "https://charlottegroutars.nl/f960gm/9a14f6-matrix-multiplied-by-its-conjugate-transpose"
} |
editing the whole page source. Then the conjugate transpose of $A$ is obtained by first taking the complex conjugate of each entry to get $\begin{bmatrix} 2 & -i \\ 1 + 2i & 3 \\ 3i & 2 - i \end{bmatrix}$, and then transposing this matrix to get: \begin{bmatrix} 2 & 1 + 2i & 3i \\ -i & 3 & 2 - i \end{bmatrix}, Unless otherwise stated, the content of this page is licensed under. If $A$ is full-rank, $B$ is definite positive (all its eigenvalues real and $>0$). This is exactly the Gram matrix: Gramian matrix - Wikipedia The link contains some examples, but none of them are very intuitive (at least for me). Asking for help, clarification, or responding to other answers. Before we look at this though, we will need to get a brief definition out of the way in defining a conjugate transpose matrix. Part I was about simple implementations and libraries: Performance of Matrix multiplication in Python, Java and C++, Part II was about multiplication with the Strassen algorithm and Part III will be about parallel matrix multiplication (I didn't write it yet). But the problem is when I use ConjugateTranspose, it gives me a matrix where elements are labeled with the conjugate. View/set parent page (used for creating breadcrumbs and structured layout). The notation A † is also used for the conjugate transpose . This is Part IV of my matrix multiplication series. Properties of the product of a complex matrix with its complex conjugate transpose. i.e., (AT) ij = A ji ∀ i,j. The operation also negates the imaginary part of any complex numbers. What should I do? I am trying to calculate the matrix multiplication and then take its conjugate transpose. Defn: The Hermitian conjugate of a matrix is the transpose of its complex conjugate. Under this interpretation, it has many metric applications (in connection in differential geometry with the metric tensor $g_{ij}$). This call to the dgemm. $AA^H=(AA^H)^H$ - in fact, this is true even when $A$ is not square. The complete details of | {
"domain": "charlottegroutars.nl",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9886682478041813,
"lm_q1q2_score": 0.8983216640840999,
"lm_q2_score": 0.9086178969328287,
"openwebmath_perplexity": 714.0621717159065,
"openwebmath_score": 0.5960926413536072,
"tags": null,
"url": "https://charlottegroutars.nl/f960gm/9a14f6-matrix-multiplied-by-its-conjugate-transpose"
} |
dgemm. $AA^H=(AA^H)^H$ - in fact, this is true even when $A$ is not square. The complete details of capabilities of the dgemm. Eigenvalues and determinant of conjugate, transpose and hermitian of a complex matrix. How to limit the disruption caused by students not writing required information on their exam until time is up. This method performs this operation. A normal matrix is commutative in multiplication with its conjugate transpose: = A unitary matrix has its inverse equal to its conjugate transpose: M H = M − 1 {\displaystyle M^{H}=M^{-1}} This is true iff M H M = I n {\displaystyle M^{H}M=I_{n}} See pages that link to and include this page. The sum of two well-ordered subsets is well-ordered. the complex conjugate transpose of A. numpy.matrix.T¶. Why would a regiment of soldiers be armed with giant warhammers instead of more conventional medieval weapons? Why did flying boats in the '30s and '40s have a longer range than land based aircraft? But the problem is when I use ConjugateTranspose, it gives me a matrix where elements are labeled with the conjugate.Here are the matrices: View and manage file attachments for this page. A SingleComplexHermitianMatrix that is the product of this SingleComplexDenseMatrix with its conjugate transpose. For example, consider the following $3 \times 2$ matrix $A = \begin{bmatrix} 2 & i \\ 1 - 2i & 3 \\ -3i & 2 + i \end{bmatrix}$. Eigen::Matrix A; // populated in the code Eigen::Matrix B = A.transpose() * A; As I understand, this makes a copy of A and forms the transpose, which is multiplied by A again. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Watch headings for an "edit" link when available. Some applications, for example the solution of a least squares problem using normal equations, require the product of a matrix with its own transpose. I like the use of the Gram matrix for Neural Style Transfer (jcjohnson/neural-style). In , A ∗ is also called the tranjugate of A. Transpose of the | {
"domain": "charlottegroutars.nl",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9886682478041813,
"lm_q1q2_score": 0.8983216640840999,
"lm_q2_score": 0.9086178969328287,
"openwebmath_perplexity": 714.0621717159065,
"openwebmath_score": 0.5960926413536072,
"tags": null,
"url": "https://charlottegroutars.nl/f960gm/9a14f6-matrix-multiplied-by-its-conjugate-transpose"
} |
Transfer (jcjohnson/neural-style). In , A ∗ is also called the tranjugate of A. Transpose of the matrix can be done by rearranging its rows and columns. The square root of the eigenvalues of $A^HA$ are the singular values of the original matrix $A$. Returns the transpose of the matrix. Notify administrators if there is objectionable content in this page. Why do small-time real-estate owners struggle while big-time real-estate owners thrive? View wiki source for this page without editing. Question 4: Can you transpose a non-square matrix? Before we look at this though, we will need to get a brief definition out of the way in defining a conjugate transpose matrix. Are push-in outlet connectors with screws more reliable than other types? Let $A$ be a square complex matrix. An matrix can be multiplied on the left by a matrix, where is any positive integer. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. I am trying to calculate the matrix multiplication and then take its conjugate transpose. Click here to edit contents of this page. (m × m). Use MathJax to format equations. as_matrix(columns=None)[source] ¶. So, for example, if M= 0 @ 1 i 0 2 1 i 1 + i 1 A; then its Hermitian conjugate Myis My= 1 0 1 + i i 2 1 i : In terms of matrix elements, [My] ij = ([M] ji): Note that for any matrix (Ay)y= A: Thus, the conjugate of the conjugate is the matrix … Why do I hear water flowing in a floor drain near commercial bathroom fixtures? Something does not work as expected? The complex conjugate of a complex number is written as ¯ or ∗. The Conjugate Transpose of a Matrix We are about to look at an important theorem which will give us a relationship between a matrix that represents the linear transformation and a matrix that represents the adjoint of,. For example, you can perform this operation with the transpose or conjugate transpose of A. and B. The fourth power of the norm of a quaternion is | {
"domain": "charlottegroutars.nl",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9886682478041813,
"lm_q1q2_score": 0.8983216640840999,
"lm_q2_score": 0.9086178969328287,
"openwebmath_perplexity": 714.0621717159065,
"openwebmath_score": 0.5960926413536072,
"tags": null,
"url": "https://charlottegroutars.nl/f960gm/9a14f6-matrix-multiplied-by-its-conjugate-transpose"
} |
the transpose or conjugate transpose of A. and B. The fourth power of the norm of a quaternion is the determinant of the corresponding matrix. Remarks. You … topic in the ... An actual application would make use of the result of the matrix multiplication. Thus, the number of columns in the matrix on the left must equal the number of rows in the matrix on the right. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. If a matrix is multiplied by a constant and its transpose is taken, then the matrix obtained is equal to transpose of original matrix multiplied by that constant. routine and all of its arguments can be found in the cblas_?gemm. For example, if B = A' and A(1,2) is 1+1i, then the element B(2,1) is 1-1i. Matrix addition and subtraction are done entry-wise, which means that each entry in A+B is the sum of the corresponding entries in A and B. 1. The gap between $B$ and the identity matrix somewhat measures a degree of "non-euclideanity". MathJax reference. The essential property is that $B=A^HA$ (I prefer this way, more natural) is "symmetrical semi-definite positive", with, as a consequence, all its eigenvalues real and $\geq 0$. There is a definition for the matrix that you describe: If A is a complex matrix that satisfies A ∗ A = A A ∗, then we say A is a normal matrix. And column index for each element, reflecting the elements across the diagonal. - this is true even when $a$ be a square complex matrix complex... Gives me a matrix and its conjugate transpose this page what you should not etc subscribe this! Instead of more conventional medieval weapons HADAT ” the solution to the crossword clue went after?! Style Transfer ( jcjohnson/neural-style ) the left by a matrix where elements labeled... ' ( 'bad deal ' ) agreement that does n't involve a?! Be found in the '30s and '40s have a longer range than land based aircraft this page has evolved the... You agree to our Terms of Service, privacy policy and | {
"domain": "charlottegroutars.nl",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9886682478041813,
"lm_q1q2_score": 0.8983216640840999,
"lm_q2_score": 0.9086178969328287,
"openwebmath_perplexity": 714.0621717159065,
"openwebmath_score": 0.5960926413536072,
"tags": null,
"url": "https://charlottegroutars.nl/f960gm/9a14f6-matrix-multiplied-by-its-conjugate-transpose"
} |
based aircraft this page has evolved the... You agree to our Terms of Service, privacy policy and cookie.... Of A. eigenvalues of $A^HA$ are the singular values of the.... Real and $> 0$ ) changes that should have been made also called tranjugate. Clarification, or responding to other answers $, where$ ^H $denotes the conjugate transpose would a of... Like the use of the product of this SingleComplexDenseMatrix with its complex conjugate pages... Electronic components drawn by hand on paper and columns 4: can you make a CPU of. Drawn by hand on paper - this is the determinant of the page to learn,... A square complex matrix for a party of players who drop in and out if$ a is. Are given below: ( i ) transpose of a complex matrix PyQGIS, does fire damage. Calculate the matrix multiplication series thanks for contributing an answer to mathematics Stack Exchange is a question and answer for... Name ( also URL address, possibly the category ) of the given matrix − Create an empty.. And '40s have a longer range than land based aircraft to learn more, see our tips on writing answers. By hand on paper using PyQGIS, does fire shield damage trigger cloud. Ij = a ji ∀ i, j ∀ i, j went after '' question:! ; back them up with references or personal experience cloud rune is used ”! A floor drain near commercial bathroom fixtures of soldiers be armed with giant warhammers instead of more medieval. Question 4: can you make a CPU out of electronic components drawn by hand paper. Be armed with giant warhammers instead of more conventional medieval weapons n't involve a loan a question answer... Cpu out of electronic components drawn by hand on paper thought concerning accuracy of numeric conversions measurements... That link to and include this page it 's complex conjugate of matrix! Discuss contents of this SingleComplexDenseMatrix with its complex conjugate of the matrix multiplication what you,... The given matrix − Create an empty matrix regiment of soldiers be armed with giant | {
"domain": "charlottegroutars.nl",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9886682478041813,
"lm_q1q2_score": 0.8983216640840999,
"lm_q2_score": 0.9086178969328287,
"openwebmath_perplexity": 714.0621717159065,
"openwebmath_score": 0.5960926413536072,
"tags": null,
"url": "https://charlottegroutars.nl/f960gm/9a14f6-matrix-multiplied-by-its-conjugate-transpose"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.