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# Assume that $(a_n)^{\infty}_{n=0}$ is a sequence given by $a_0=1$ and $a_{n+1}=2a_{n}+n$ for all $n \ge 0$, what is the formula for $a_n$?
Assume that $(a_n)^{\infty}_{n=0}$ is a sequence given by $a_0=1$ and $a_{n+1}=2a_{n}+n$ for all $n \ge 0$, what is the formula for $a_n$?
we know that $F(x)=a_0+a_1x+a_2x^2+a_3x^3+\dots$
and our sequence is $1, 2, 5, 12, 27, 58, \dots$
to find the generating function for the sequence I started with
$F(x)=a_0+(\underline{2a_0}+\widetilde0)x+(\underline{2a_1}+\widetilde1)x^2+(\underline{2a_2}+\widetilde2)x^3+\dots$
after distributing the variables into the parenthesis, i separated the terms with $\underline{}$ and $\widetilde{}$
$\underline{} = 2a_0x+2a_1x^2+2a_2x^3+2a_3x^4+\dots$
$\underline{} = 2x(a_0+a_1x^1+a_2x^2+a_3x^3+\dots)$
$\underline{} = 2x \times F(x)$
$\widetilde{} = x^2 + 2x^3 + 3x^4 + 4x^5$
now I don't know what to do with this part, if I had used $n$ earlier instead of putting in the numbers I could just factor out $nx^2$ and get $\widetilde{} = nx^2(\frac{1}{1-x})$ but I don't think I can have $n$ when solving for the generating function of the sequence?
Any ideas of how to get past just this part?
• Pull an $x^2$ out and note that $$\frac{d}{dx}\left(\frac{1}{1-x}\right)=1+2x+3x^2+4x^3+...$$ Oct 22, 2016 at 1:56
• @Eleven-Eleven If I use your idea then I would get $F(x)=1+2xF(x)+\frac{1}{(1-x)^2}$, this equals $F(x)=\frac{(1-x)^2+1}{(1-2x)(1-x)^2}$ then I could use partial fractions to separate this into three terms with a $1-\underline{}x$ in the denominators, find a summation and find $a_n$ Oct 22, 2016 at 2:21
• Yes except you have $x^2$ in the numerator, not 1. Oct 22, 2016 at 2:27
• For all things generating functions, look up Wilf's Generatingfuntionology... it is free to download online and you get all the tricks to solve such expression... Oct 22, 2016 at 2:32 | {
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From the recursion formula, you can guess a linear combination of $a_n$ and $n$ will be a geometric sequence, that is, you can assume $a_{n+1}-\lambda(n+1)+M=2(a_n-\lambda n+M)$ for some $\lambda$ and $M$. Put it into the the recursion formula, we can take $\lambda=-1$ and $M=1$. So we get $$a_{n+1}+(n+1)+1=2(a_n+n+1).$$
Let $b_n=a_n+n+1$, then $b_{n+1}=2b_n$, which implies that $b_n=2^nb_0$. Since $b_0=a_0+0+1=2$, then $b_n=2^{n+1}$, which implies that $$a_n=b_n-n-1=2^{n+1}-n-1.$$
• Your method works. However, the op is trying to use generating functions to solve... Oct 22, 2016 at 2:02
My preferred approach using generating functions starts by assuming that $a_n=0$ for $n<0$ and rewriting the recurrence so that it hold for all $n\ge 0$:
$$a_n=2a_{n-1}+n-1+2[n=0]\;.\tag{1}$$
Here $[n=0]$ is an Iverson bracket, and the last term of $(1)$ ensures that $a_0=1$. Now multiply $(1)$ by $x^n$ and sum over $n\ge 0$; if we let $g(x)=\sum_{n\ge 0}a_nx^n$ be the generating function, we have
\begin{align*} g(x)&=\sum_{n\ge 0}a_nx^n\\ &=2\sum_{n\ge 0}a_{n-1}x^n+\sum_{n\ge 0}nx^n-\sum_{n\ge 0}x^n+2\\ &=2x\sum_{n\ge 0}a_nx^n+x\sum_{n\ge 0}nx^{n-1}-\frac1{1-x}+2\\ &=2xg(x)+\frac{x}{(1-x)^2}-\frac1{1-x}+2\;. \end{align*}
Solving for $g(x)$, decomposing the resulting rational function into partial fractions, and rewriting the resulting terms as formal power series, we get
\begin{align*} g(x)&=\frac{x}{(1-x)^2(1-2x)}-\frac1{(1-x)(1-2x)}+\frac2{1-2x}\\ &=\frac{x-(1-x)+2(1-x)^2}{(1-x)^2(1-2x)}\\ &=\frac{1-2x+2x^2}{(1-x)^2(1-2x)}\\ &=\frac2{1-2x}-\frac1{(1-x)^2}\\ &=2\sum_{n\ge 0}(2x)^n-\sum_{n\ge 0}(n+1)x^n\\ &=\sum_{n\ge 0}\left(2^{n+1}-n-1\right)x^n\;, \end{align*}
which yields the closed form
$$a_n=2^{n+1}-n-1\;.$$
You must note that $a_0=1$ which is unaccounted for...
Using the fact that
$$\frac{d}{dx}\left(\frac{1}{1-x}\right)=\frac{1}{(1-x)^2}=1+2x+3x^2+...$$
you now should get | {
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$$\frac{d}{dx}\left(\frac{1}{1-x}\right)=\frac{1}{(1-x)^2}=1+2x+3x^2+...$$
you now should get
$$F(x)=1+2xF(x)+x^2\frac{1}{(1-x)^2}$$ $$F(x)(1-2x)=1+\frac{x^2}{(1-x)^2}$$ $$F(x)=\frac{1-2x+2x^2}{(1-2x)(1-x)^2}$$
Now use partial fraction decomposition to obtain three power series and look at the coefficients.
• just making sure, it's probably a typo but shouldn't the numerator be $1-2x+2x^2$? Oct 22, 2016 at 2:34
• Yes.... on an iPhone so mistakes are bound... Oct 22, 2016 at 2:36 | {
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# Speed of two trains travelling side by side
I'm a high school student, and I have come across a problem that I cannot solve. I feel there must be something obvious that I'm not seeing.
Problem: The distance between two train stations is $96$ km. One train covers this distance in $40$ minutes less time than another one. The second train is $12$ km/h faster than the first one. Find both trains' speeds.
What I have done: Set $v_1+12 = v_2$ (the speed of train $2$ is $12$km/h more than speed of train $2$), and $96/(v_1) = (96/v_2)-40$ (the time it takes for train 2 to transverse the distance between the stations is $40$ minutes less than the required by train $2$) Now, from here I get to: $v_1 = v_2-12$.
\begin{align} &\frac{96}{v_2-12} = \frac{96}{v_2}-40 \\ &\qquad\implies \frac{96}{v_2-12} = \frac{96-40v_2}{v_2} \\ &\qquad\implies v_2\cdot 96 = (v_2-12)\cdot (96-40v_2) \\ &\qquad\implies v_2\cdot 96 = v_2\cdot 96-40v_2^2-1152-380v_2 \\ &\qquad\implies 0 = -40v_2^2-380v_2-1152 \end{align}
Solving this quadratic equation yields no real roots.
Could you please suggest the right way to go? | {
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Solving this quadratic equation yields no real roots.
Could you please suggest the right way to go?
• You are not going to get a sensible solution unless the faster train takes less time – Henry Jan 10 '18 at 14:34
• You wrote this: $96/(v_1)=(96/v_2)−40$ (the time it takes for train 2 to transverse the distance between the stations is $40$ minutes less than the required by train 2). The equation and the parenthetical remark contradict each other. If your equation is correct, $96/(v_1)$ is smaller ($40$ less) than $96/(v_2)$, so your equation says train 1 is faster. – Steve Kass Jan 10 '18 at 14:39
• +1 for first time poster a.) showing work, b.) not flat out asking for the answer – jameselmore Jan 10 '18 at 14:40
• If you find one or more of the answers helpful, it is customary to "Accept" one of them. Either accept the answer that you think is most helpful, or (if there are several that are all just as good), you might consider accepting the answer from the user with the least reputation (as accepting an answer gives a small reputation reward). – Xander Henderson Jan 10 '18 at 15:53
• The moral of the story is: write down your units! Now that you have found your mistake, here is a follow up problem. The two trains are at opposite ends of the 96km route, heading towards each other on the same track at the speeds you have deduced. How far apart are they when they collide? – Eric Lippert Jan 10 '18 at 19:53 | {
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Let $v_1$ km/hr denote the speed of the faster train, and $v_2$ km/hr denote the speed of the slower train (I seem to have reversed your notation—sorry, $v_1$ feels like faster variable to me than $v_2$). First off, we can relate the amount of time it takes for each train to travel the 96 km to the speed of each train. So, let $$t_1 \text{ hrs} = \frac{96 \text{ km}}{v_1 \ \frac{\text{km}}{\text{hr}}} = \frac{96}{v_1}\text{ hrs} \qquad\text{and}\qquad t_2 \text{ hrs} = \frac{96 \text{ km}}{v_2 \ \frac{\text{km}}{\text{hr}}} = \frac{96}{v_2}\text{ hrs}\tag{1}$$ denote these two times. We know that the faster train arrives 40 minutes (that is, $\frac{2}{3}$ of an hour—watch the units! (this is an easy mistake to make—I messed it up, too!)) earlier than the slower train, which implies that $$t_1 \text{ hrs} = t_2 \text{ hrs} - \frac{2}{3} \text{ hrs} = \left( t_2 - \frac{2}{3} \right)\text{ hrs},$$ and we know that the faster train is 12 kph faster than the slower train, hence $$v_1 \ \frac{\text{km}}{\text{hr}} = v_2 \ \frac{\text{km}}{\text{hr}} + 12 \ \frac{\text{km}}{\text{hr}} = \left(v_2 + 12\right) \ \frac{\text{km}}{\text{hr}}.$$ It should be noted that the only major mistake that I see in your work is in the above step—in your model the faster train takes more time to cover the distance, which is a problem. Substituting these into the equations at (1) (and eliding units—the units of time are hours, the unit of distance are kilometers, and the units of speed are km/hr), we get the system $$\begin{cases} t_2 - \dfrac{2}{3} = \dfrac{96}{v_2 + 12} \\ t_2 = \dfrac{96}{v_2}. \end{cases}$$ Can you solve it from here? | {
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• Thanks to everyone!! First, I had messed up my vars as Henderson, SteveKass and others pointed, and second, I was ignoring the minute - hour relaationship. Solved. Lots of thanks. – Vladislav Vordank Jan 10 '18 at 15:25
• Xander, I wonder if you would consider actually including the units in your equations? I consider writing units to be such an important best practice (and, in particular, one that would have made one of the errors obvious) that I'd like to see it represented among the answers. But I don't think I could improve on your explanation, which is why I didn't just write my own answer. – David Z Jan 11 '18 at 0:38
• @DavidZ Indeed, and I probably would have done that in the first place if the "obvious" sign error hadn't made me complacent. I've edited the answer to include units in the setup. – Xander Henderson Jan 11 '18 at 6:29
• Thanks! (Your edit looked a little strange at first and I took a while to realize it's because I would omit units after the variables, e.g. let $v_1$ represent a speed, not just a number, and then write things like $v_1 = v_2 + 12\ \mathrm{km/hr.}$, but I suppose that's just a matter of preference.) – David Z Jan 11 '18 at 7:08
The question states that "One train covers this distance in 40 mins less than the other". Although it does not tell you which train, it is quite obvious that the faster train (i.e. train 2) takes 40 mins less.
So instead, it should be $96/(v2)=(96/v1)−40$.
• Also, you've mixed up hours and minutes. It should be 2/3 not 40. – Michael Behrend Jan 10 '18 at 14:46
• Thanks to everyone!! First, I had messed up my vars as Henderson, SteveKass and others pointed, and second, I was ignoring the minute - hour relaationship. Solved. Lots of thanks. – Vladislav Vordank Jan 10 '18 at 15:25
I'll post this as an answer, since i'm not yet allowed to write comments. So, as mentioned in the comments (and the answer given by glowstonetrees) you might want to use $96/(v2)=(96/v1)−40$ instead. | {
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Also keep in mind that you are subtracting minutes from hours. Your final formula should be $96/(v2)=(96/v1)−(40/60)$.
• Thanks to everyone!! First, I had messed up my vars as Henderson, SteveKass and others pointed, and second, I was ignoring the minute - hour relaationship. Solved. Lots of thanks. – Vladislav Vordank Jan 10 '18 at 15:25
A method that doesn't require the quadratic formula:
Once you've corrected the errors mentioned in the other answers, you should have
96/(v+12) = 96/v – 2/3
You can rewrite that as
8/(v/12 +1) = 8/(v/12) -2/3
12/(v/12+1)=12/(v/12) – 1
This is in some ways a more complicated form, but if we assume that each term is an integer, then v/12 and v/12+1 must be factors of 12. And what factors of twelve are there that differ by 1? Just 3,4. If we plug those in and check, 12/4 = 12/3 -1 => 3 = 4-1. So v/12 = 3 => v = 36 and the other speed is 4*12 = 48.
Also, when you get an equation like
v2⋅96=(v2−12)⋅(96−40v2)
You should divide out by the common factor of 8 and get
v2⋅12=(v2−12)⋅(12−5v2)
before multiplying it out. | {
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# The approximate uncertainty in $r$
The volume of a cylinder is given by the expression
$V=\pi r^2 h$
The uncertainties for $V$ and $h$ are as shown below
\begin{align} V&\pm 7\%\\ h&\pm 3\% \end{align}
What is the uncertainty in $r$?
Now, the obvious answer would be $2\%$, from $$\frac{dV}{V}=\frac{dh}{h}+2\frac{dr}{r}$$
However, rearranging to $r^2=\frac{V}{\pi h}$ gives $$2\frac{dr}{r}=\frac{dV}{V}+\frac{dh}{h}$$ which gives a different answer of $5\%$. Thus, by simply rearranging the formula, we get different values for uncertainty in $r$.
How do you explain this?
(The mark scheme lists $5\%$ as the correct answer)
-
@Helder, thats fine! – 0sh Sep 22 '11 at 13:09
You're confusing independent and dependent variables. When you propogate from uncertainties in the $x_{i}$ to some $f(x_{1},x_{2}...)$, the formula $\delta f(x_{1}...)=\sum \left|\frac{\partial f}{\partial x_{i}}\right|\delta x_{i}$ assumes that each of the $x_{i}$ is an independently measured variable and that $f$ is a dependent variable to be calculated from the $x_{i}$.
In the example you give, you have two independent measurements of $V$ and $h$ and are expected to calculate the uncertainty in $r$. Well, to use the above formula, you need to write $r$ as a dependent variable of $V$ and $h$. Therefore, it's only correct to solve for $r$ first, and then calculate the uncertainty.
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Makes sense, thanks! – 0sh Apr 20 '11 at 12:36
Ted Bunn's answer is a very important addition to this answer. – Qmechanic Apr 20 '11 at 17:56
@Qmechanic: agreed. I typed this up to get the jist of an answer before I had to leave for work. – Jerry Schirmer Apr 20 '11 at 18:26
Jerry Schirmer's right about why solving for $r$ first is the right procedure. One way to illustrate this is to notice that with the other procedure the uncertainty could go negative, which can't be right. | {
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But the main thing I wanted to point out is that, if the measurements of $V$ and $h$ are independent, and if the "errors" mean standard deviations as usual, then the correct procedure is to add the errors in quadrature (i.e., to add the squares and take the square root): $${\delta r\over r}={1\over 2}\sqrt{\left(\delta V\over V\right)^2+\left(\delta h\over h\right)^2}.$$ See, e.g., any of the first few Google hits for "propagation of errors."
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the measurements of V and h are independent? – Helder Velez Apr 26 '11 at 2:35
That's what one is apparently meant to assume in this problem. Without more detail about how the measurements were made, there's no way to tell if this is reasonable, but if $h$ is measured with a ruler and $V$ is measured by dunking the thing in water and watching the water level rise, for instance, this would be correct. If $V$ is measured by measuring $r$ and $h$ and calculating $\pi r^2h$, then of course the measurements wouldn't be independent, but the question would make no sense if that were the case, so I think we're meant to assume it's not. – Ted Bunn Apr 26 '11 at 2:52
This should be a comment to Jerry Schirmer's right answer.
The solutions you give solve different problems.
In the first calculation you measure V with its uncertainty $\Delta V$, you know it depends with r and h, you know the uncertainty you have in the measurement of h, $\Delta h$ and thus you infer the uncertainty you had associated with measuring r, $\Delta r$.
In the second calculation you want to measure r from V and h. Knowing the uncertainties in V and h you predict the uncertainty that you will have in r.
-
Your first formula is correct, but it looks like you substituted in the incorrectly signed value for (dh/h). Your second formula is missing a sign change for the (dh/h) term.
In the first equation, substituting, to get the MAXIMUM (dr/r) --->
+7% = (-3%) + (2 x (+5%)) -OR- -7% = (+3%) + (2 x (-5%)) | {
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+7% = (-3%) + (2 x (+5%)) -OR- -7% = (+3%) + (2 x (-5%))
So, your text is correct: +/-5% is the (approximate) uncertainty for r.
If you solve this problem algebraically, you will find that the (corrected) differential equations you constructed gives a very close but not exact answer. The correct uncertainty for r is something like + SQRT(1.07/0.97) and - SQRT(0.93/1.03) = +5.0282% / -4.9783%. This is because the differential eqns are valid only for infinitesimal deltas; but we are using deltas on the order of +/-5% (hardly infinitesimal).
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I will use two ways to derive the uncertainity on r
1 - very simple method: the 'interval arithmetic' present in the Euler math package (free)
2 - derive the formula of dr(V,h, dV,dh)
Both answers : 5% (for any value V or h that I choose for V and h)
// 'using interval arithmetic'
>dV=0.07 dh=0.03 // deltas
>V=1000 h=40 //arbitrary choosen values
>r0=sqrt(1/pi*V/h) //central value for r
2.82094791774
>r=sqrt(1/pi*((V±V*dV)/(h±h*dh))) //interval values for r
~2.6,3~ // from 2.6 to 3
>(r-r0)
~-0.14,0.14~
>(r-r0)/r0
**~-0.05,0.05~ // 5% for dr/r0 for any V, h**
//calc dr(V,h, dV,dh) (seeResistance_measurement Example and adapting the derivatives)
dr=1/2/sqrt(pi)*sqrt(dV^2/V/h+dh^2*V/h^3)
0.00106245302667
1-(r-dr)/r0
~-0.05,0.05~ // the result is again 5%
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Summation of series 12,40,90,168,280,432,…? [closed]
I have been stuck in finding the general term and the sum of $n$ terms of the series
$$12,\,40,\,90,\,168,\,280,\,432, ...$$
I am not able to see any relationship between the successive terms of the series.
Is there a pattern which I am not able to see?
• There has to be. If there is no pattern, you can't come up with the other terms. – Laray Jan 17 '17 at 16:07
• There is a pattern, which clearly points at the next term being $$42$$ as every other term after it. Thus, for every $n\geqslant6$, the sum of the $n$ first terms is $$42n+770$$ – Did Jan 17 '17 at 16:12
• If you compute repeated differences, then you'll see that there is a simple cubic polynomials that explains these values: WA. – lhf Jan 17 '17 at 16:13
Clearly there is a pattern.
Firstly find the orders of differences.
They are
28, 50, 78, 112, 152,...
22, 28, 34, 40, ....
6, 6, 6, ....
0, 0,.....
Hence, the $n^{th}$ term can be written as
$12+28(n-1)+\frac{22n(n-1)(n-2)}{2!}+\frac{6n(n-1)(n-2)(n-3)}{3!}$
=$n^3+5n^2+6n$
The sum of n terms of the above series is
$\sum {n^3}+5 \sum {n^2}+6\sum {n}$
=$12n$+$\frac{28n(n-1)}{2!}+\frac{22n(n-1)(n-2)}{3!}+\frac{6n(n-1)(n-2)(n-3)}{4!}$
=$\frac{n(3n^2+26n+69n+46}{12})$
=$\frac{1}{12}n(n+1)(3n^2+23n+46)$
There is indeed. Observe that if you go on taking successive differences (at each step) then, you get
$$12,40,90,168,280,432,...$$ $$28,50,78,112,152,...$$ $$22,28,34,40,...$$ $$6,6,6,6,...$$
Perhaps, a constant sequence. Does this strike something?
Yes! You can assert that, the general term of the given sequence is of the form $x_n = an^3+bn^2+cn+d$. Solve for $(a,b,c,d)$ using the fact that $x_1=12, x_2 = 40, x_3 = 90$ and $x_4 = 168$.
The Online Encyclopedia of Integer Sequences thinks that the terms are given by $$a_n=n^3+5n^2+6n$$ | {
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The Online Encyclopedia of Integer Sequences thinks that the terms are given by $$a_n=n^3+5n^2+6n$$
• So does WA. – lhf Jan 17 '17 at 16:14
• Divide the entries by $2$ and you get oeis.org/A005564 – Barry Cipra Jan 17 '17 at 16:24
Per my solution here it can be shown that the $n$-th term ($n=0,1,2,3,\cdots$), $b_n$ is given by
$$b_n=\sum_{r=0}^{\min(3,n)}\binom nr a_r$$ where $a_r=12,28,22,6$ for $r=0,1,2,3$, i.e.
\begin{align} b_0&=\binom 00 12&&=12\\ b_1&=\binom 10 12+\binom 11 28&&=40\\ b_2&=\binom 20 12+\binom 21 28+\binom 22 22 &&=90\\ b_3&=\binom 30 12+\binom 31 28+\binom 32 22 +\binom 33 6&&=168\\ b_4&=\binom 40 12+\binom 41 28+\binom 42 22 +\binom 43 6&&=280\\ b_5&=\binom 50 12+\binom 51 28+\binom 52 22 +\binom 53 6&&=432\\ \vdots\\ \color{red}{b_n}&\color{red}{=\binom n0 12+\binom n1 28 + \binom n2 22 +\binom n3 6}\\ &\color{red}{=n^3+8n^2+19n+12}\\ &\color{red}{=(n+1)(n+3)(n+4)} \end{align}
Note that $a_r$ is the first term of the $r$-th difference series, with $r=0$ referring to the original series.
The sum of the first $n$ terms is given by
\begin{align} S_n &=\sum_{r=0}^n \binom r0 12 +\underbrace{\sum_{r=1}^n\binom r1 28 + \underbrace{\sum_{r=2}^n\binom r2 22 +\underbrace{\sum_{r=3}^n\binom r3 6}_{n\ge 3}}_{n\ge 2}}_{n\ge 1}\\\\ &=\color{red}{\binom {n+1}1 12 +\underbrace{\binom {n+1}2 28 +\underbrace{\binom {n+1}3 22 +\underbrace{\binom {n+1}4 6}_{n\ge 3}}_{n\ge 2}}_{n\ge 1}}\\ &\color{red}{=(n+1)(3n^3+13n^2-2n+12)\qquad [\text{for }n\ge 3]} \end{align}
NB: References above to the $n$-th term count from $n=0$. | {
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# What is the probability that both cards are not aces?
Suppose two cards are drawn from a standard 52 card deck without replacement. Assuming all cards are equally likely to be selected, what is the probability that both cards are not aces?
My Solution
A = Event that first card is an ace
B = Event that second card is an ace given that first is an ace
C = Event that both cards are aces
D = Event that both cards are not aces $$P(A) = \frac{4}{52}$$ $$P(B) = \frac{3}{51}$$ $$P(C) = \frac{4}{52}*\frac{3}{51} = \frac{1}{221}$$ $$P(D) = 1 - P(C) = 1 - \frac{1}{221} = \frac{220}{221}$$
Actual Solution
A = Event that first card is not an ace
B = Event that second card is not an ace given that first is not an ace
C = Event that both cards are not aces $$P(A) = \frac{48}{52}$$ $$P(B) = \frac{47}{51}$$ $$P(C) = \frac{48}{52}*\frac{47}{51} = \frac{188}{221}$$
Why my solution that P(D) = 1 - P(C) is wrong? | {
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Why my solution that P(D) = 1 - P(C) is wrong?
• Somewhat ambiguous question. Suppose the two drawn cards are called A and B. Do you mean A is not an ace and B is also not an ace? It could be interpreted as A and B are both not aces (as a pair) but one of them could be an ace. As with all good math word problems, it is important to understand the question clearly first before attempting to solve it. – David Jul 20 '16 at 11:18
• @David I am now confused with my question after seeing your comment. But the actual answer states its 188/221 so I assume A is not an ace and B is also not an ace. – jblixr Jul 20 '16 at 14:50
• Yes that seems to be the correct interpretation that neither of the $2$ drawn cards are aces so you drew $0$ aces. An "easy" way to compute this probability is imagine we remove all $4$ aces from the deck of $52$, leaving us with $48$ cards only. We have $48$ ways to draw the first card and $47$ ways to draw the 2nd card. Now imagine we put the deck back to $52$ cards and draw $2$ cards. We have $52$ ways to draw the 1st card and $51$ ways to draw the 2nd card. Therefore, the probability of drawing no aces for the first $2$ cards is $(48*47)/(52*51)$. – David Jul 21 '16 at 3:20
• In the above comment, we can ignore the "missing" divide by $2$ for both the numerator and the denominator since they will cancel out. Conceptually, this is about the simplest way I can explain the answer. That is, remove the aces to guarantee no aces drawn the draw $2$ non ace cards which is exactly what we want. Then put all the aces back and draw $2$ cards . Then compute the ratio and that is the probability (about $85$%). – David Jul 21 '16 at 3:45
The probability that both cards are not aces is the complement of the event that at least one of the cards selected is an ace. You overlooked the possibility that exactly one of the two cards selected is an ace. | {
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The probability that both cards are aces is $$\frac{4}{52} \cdot \frac{3}{51} = \frac{1}{221}$$
The probability that the first card is an ace and the second card is not an ace is $$\frac{4}{52} \cdot \frac{48}{51} = \frac{16}{221}$$ The probability that the first cards is not an ace and the second card is an ace is $$\frac{48}{52} \cdot \frac{4}{51} = \frac{16}{221}$$ Therefore, the probability that at least one of the two cards is an ace is $$\frac{1}{221} + \frac{16}{221} + \frac{16}{221} = \frac{33}{221}$$ Hence, the probability that both cards are not aces is $$1 - \frac{33}{221} = \frac{188}{221}$$ which can be found more simply by using the method you included in your post.
This may be a language issue. You calculated the probability that not both cards are aces, whereas the problem asks for the probability that both cards are not aces.
Since you overloaded your event variables, I'll define new ones: Let $E$ be the event that the first card is an ace and $F$ the event that the second card is an ace; then the event that not both cards are aces is $\overline{E\cap F}=\overline E\cup\overline F$, and the event that both cards are not aces is $\overline E\cap\overline F=\overline{E\cup F}$.
If you want neither of the first $2$ drawn cards to be aces then it is simply $48 \choose 2$ / $52 \choose 2$ which is $188/221$. Here we are simply choosing $2$ cards from the $48$ non aces and are dividing by the total number of possible $2$ card pairs from the full deck of $52$.
If $C$ is the event that both cards are aces, then $1 - P(C)$ is the probability that at least one of the two cards is not an ace.
What you need to be calculating (using your A,B) is:
$$(1-P(A))(1-P(B))$$
which is the probability that the first card is not an ace, and the second is an ace given that the first is not. | {
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# Show that $\sum\limits_{n\ge1}\frac1{n^2}=\sum\limits_{n\ge1}\frac3{n^2\binom{2n}n}$ without actually evaluating both series
$$\sum\limits_{n\ge1}\frac1{n^2}=\sum\limits_{n\ge1}\frac3{n^2\binom{2n}n}\tag1$$
Note that $$(1)$$ holds since the LHS is given by $$\zeta(2)$$ whereas the RHS by $$6\arcsin^2 1$$ which both equal $$\dfrac{\pi^2}6$$ as it is well-known. However I am interested in proving $$(1)$$ without actually evaluating both series.
I am aware of an elegant approach contributed by Markus Scheuer as an answer to Different methods to compute Basel problem. Although this answers my question partially I am looking for different attemps. For example within Jack D'Aurizio's notes there is a way proposed exploiting creative telescoping $$($$see page $$5$$f.$$)$$ which I am sadly speaking not able to understand completely yet. Hence I have come across a proof of a similiar equality concerning $$\zeta(3)$$ on AoPS given by pprime I am confident that there are in fact other possible methods.
I would like to see attempts of proving $$(1)$$ beside the mentioned which do not rely on actually showing that they both equal $$\dfrac{\pi^2}6$$. Preferably these should be in the spirit of Markus Scheuer's or Jack D'Aurizio's approaches rather than the one similiar by pprime.
EDIT I
I have found another interesting approach, again by Jack D'Aurizio, which can be found here utilizing harmonic sums and creative telescoping in combination.
EDIT II
As pointed out by Zacky on page $$31$$ of Jack's notes another methode can be found which makes it three possibilities provided by Jack alone. Quite impressive! | {
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• Here is a similar type of series (just 2 hours ago, also answered by yourself), using $\arcsin$. But yes, here is the series evaluated directly. Seems pretty good to me. – Dietrich Burde Dec 24 '18 at 12:53
• @DietrichBurde I am not sure how this is of use. Could you elaborate on the utility of the linked post in order to answer my question? Furthermore I am aware of this post since I posted an answer there $2$ hours ago ^^ – mrtaurho Dec 24 '18 at 12:59
• I thought, it would be the best to show that both sides are equal to $\zeta(2)$, but you want a different solution (and I just do not know why this should be better or more interesting, but this is of course due to my missing understanding). – Dietrich Burde Dec 24 '18 at 13:08
• @DietrichBurde Ah okay. I have not considered it from this point of view. Regarding to the value of alternative proofs: I am just interested in different approaches. For sure it is quite convincing to show both series equal the same value but out of experience I have observed that is most likely to be a way more difficult to show the equality of the series all by themselves $($see for example here: math.stackexchange.com/q/2942630 where an elegant trick was needed to show that the integrals are equivalent$)$. – mrtaurho Dec 24 '18 at 13:10 | {
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Since we have $$\frac{1}{\binom{2n}{n}} =\frac{n!n!}{(2n)!}$$ by the binomial identity, we obtain $$\frac{1}{n^2\binom{2n}{n}} =\frac{(n-1)!(n-1)!}{(2n)!} =\frac{\color{purple}{\Gamma(n)\Gamma(n)}}{2n\color{purple}{\Gamma(n+n)}}=\frac{1}{2n}\color{purple}{B(n,n)}$$ Therefore we get \begin{align*} \sum_{n=1}^\infty \frac{1}{n^2\binom{2n}{n}} & =\frac12\sum_{n=1}^\infty \frac{1}{n}\color{purple}{B(n,n)} =\frac12\sum_{n=1}^\infty \frac{1}{n}\color{purple}{\int_0^1 (x(1-x))^{n-1}dx} \\ & = -\frac12\int_0^1 \frac{1}{x(1-x)} \color{blue}{\left(-\sum_{n=1}^\infty \frac{(x(1-x))^{n}}{n}\right)}dx = - \frac12 \int_0^1 \frac{\color{blue}{\ln(1-x(1-x))}}{x(1-x)}dx \\ & = -\frac12 \bigg(\int_0^1 \frac{\ln(1-x(1-x))}{x}dx+\underbrace{\int_0^1 \frac{\ln(1-x(1-x))}{1-x}dx}_{1-x\ \rightarrow \ x}\bigg) \\ & = - \frac12\left(\int_0^1 \frac{\ln(1-x(1-x))}{x}dx +\int_0^1 \frac{\ln(1-(1-x)x)}{x}dx\right) \\ & = - \int_0^1 \frac{\ln(1-x+x^2)}{x}dx = - \int_0^1 \frac{\ln\left(\frac{1+x^3}{1+x}\right)}{x}dx \\ & = \int_0^1 \frac{\ln(1+x)}{x}dx-\underbrace{\int_0^1 \frac{\ln(1+x^3)}{x}dx}_{x^3 \rightarrow x} \\ & = \int_0^1 \frac{\ln(1+x)}{x}dx -\frac13 \int_0^1 \frac{\ln(1+x)}{x}dx =\frac23\int_0^1 \frac{\ln(1+x)}{x}dx \end{align*} $$\quad \quad \quad \quad \quad \quad \displaystyle{ =\frac13 \int_0^1 \frac{\ln x}{x-1}dx}$$$$\displaystyle{=-\frac13\sum_{n=0}^\infty \int_0^1 x^n \ln xdx= \frac13 \sum_{n=0}^\infty \frac{1}{(n+1)^2}=\frac13 \sum_{n=1}^\infty \frac{1}{n^2}}$$
As an alternative just take $$x=1$$ in the following relation shown by Felix Marin: $$\sum_{n = 1}^{\infty}{x^{n} \over n^{2}{2n \choose n}} =-\int_{0}^{1} \frac{\ln(1-(1-t)tx)}{t} dt.$$ | {
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# Solve algebraically: $\lim\limits_{x \to 3} \frac{3-x}{5-\sqrt{x^2+16}}$
$$\lim\limits_{x \to 3} \frac{3-x}{5-\sqrt{x^2+16}}$$
The professor says we can't use l'hopital's rule and must solve algebraically.
• Multiply top and bottom by $5+\sqrt{x^2+16}$. Then factor the resulting polynomial downstairs and cancel what you can. – David Mitra Feb 28 '14 at 15:17
## 2 Answers
Try multiplying the numerator and denominator by the conjugate of the denominator:
Multiply by $$\dfrac{5+ \sqrt{x^2 +16}}{5+\sqrt{x^2 + 16}}$$
$$\lim\limits_{x \to 3} \frac{3-x}{5-\sqrt{x^2+16}} \cdot \dfrac{5+ \sqrt{x^2 +16}}{5+\sqrt{x^2 + 16}} = \lim_{x\to 3}\dfrac{(3-x)(5 + \sqrt {x^2 + 16)}}{25 - (x^2 + 16)}$$
Then note that you have a difference of squares in the denominator: $$25 - (x^2 + 16) = 9 - x^2 = (3-x)(3+x)$$
Now, you can cancel the factor $3-x$, as it appears in both the numerator and denominator.
• Thank you so much for you quick reply! I understand and found the limit of 5/3! – dev Feb 28 '14 at 15:28
• Exactly! Nice work, dev. – Namaste Feb 28 '14 at 15:28
Hint $\ \ f\bar f = (a\!-\!x)(a\!+\!x)\ \Rightarrow\ \dfrac{a-x}f \,=\, \dfrac{\bar f}{a+x}\ \$ where $\ \ \bar f,\,f\, =\, 5\pm \sqrt{x^2+16}$
Remark $\$ This is a special case of rationalizing the denominator, which is often handy.
• Does that link imply those three accounts all belong to you? – TMM Feb 28 '14 at 15:49 | {
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# Category: Mathematics
## MCQs Mathematics 2
The GRE mathematical reasoning section tests your knowledge of arithmetic, basic algebra, applied mathematics, elementary geometry, and common graphs and charts. Let Starts with the GRE MCQs Mathematics quiz.
1. If the value of a piece of the property decreases by 10% while the tax rate on the property increases by 10%, what is the effect on the taxes?
2. In a cafeteria, the people are either faculty members or students. The number of faculty members is 15% of the total number of people in the cafeteria. After some of the students leave, the total number of persons remaining in the cafeteria is 50% of the original total. The number of students who left is what fractional part of the original number of students?
3. If $x=\frac{y}{7}$ and $7x=12$, then $y=?$
4. ABC works two part-time jobs. One week ABC worked 8 hours at one job, earning $\$150$, and$4.5$hours at the other job, earning$\$90$. What were his average hourly earnings for the week?
5. $2000 amount is deposited into a savings account that earns interest at the rate of 10% per year, compounded semi-annually. How much money will there be in the account at the end of one year? 6. A certain photo state machine produces 13 copies every 10 seconds. If the machine operates without interruption, how many copies will it produce in an hour? 7. If$3x-5=x+11$, then$x=$? 8. If$x=k+\frac{1}{2}=\frac{k+1}{2}$, then$x=$? 9. If$x+y=8$and$2x-y=10$then$x=$? 10. In a certain population, 40% of all people have biological characteristics$X$; the others do not. if 8000 people have characteristic$X$, how many people do not have$X$? 11. For how many 3-digit whole numbers is the sum of the digits equal to 3? 12. A car dealer who gives a customer a 20% discount on the list price of a car still realizes a net profit of 25% of the cost. If the dealer’s cost is$4800 what is the usual list price of the car?
13. If $7-x=0$, then $10-x=$?
GRE Mathematics-1
## GRE Mathematics 1 | {
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13. If $7-x=0$, then $10-x=$?
GRE Mathematics-1
## GRE Mathematics 1
The GRE mathematical reasoning section tests your knowledge of arithmetic, basic algebra, applied mathematics, elementary geometry, and common graphs and charts.
Please go to GRE Mathematics 1 to view the test
The problem solving questions are typically word problem questions. These sort of questions are usually part of different competitive and ability tests.
Take another test: Sequence and Series | {
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mathematically, SEM = SD/√(sample size). Everybody with basic statistical knowledge should understand the differences between the standard deviation (SD) and the standard error of mean (SE or SEM). A t-test is a statistical method used to see if two sets of data are significantly different. x When the true underlying distribution is known to be Gaussian, although with unknown σ, then the resulting estimated distribution follows the Student t-distribution. We are an Essay Writing Company Get an essay written for you for as low as $13/page simply by clicking the Place Order button! It has a great role to play the testing of statistical hypothesis and interval estimation. Standard deviation tells you how spread out the data is. {\displaystyle n} If a number is added to a set that is far away from the mean, how does this affect standard deviation? How can you calculate the Confidence Interval (CI) for a mean? It gives an idea of the exactness and … such that. ), you need to compare it to your estimate of the population mean and your estimate of the population standard deviation (not the sample mean's standard deviation, also known as SEM). {\displaystyle \sigma _{\bar {x}}} Where: s = sample standard deviation x 1, ..., x N = the sample data set x̄. a statistical index of the probability that a given sample mean is representative of the mean of the population from which the sample was drawn. Access this article for 1 day for:£30 /$37 / €33 (excludes VAT). 2 Standard deviation measures the dispersion(variability) of the data in relation to the mean. NOTE: We only request your email address so that the person you are recommending the page to knows that you wanted them to see it, and that it is not junk mail. Regression line deviation for more discussion if any, are true away from the estimate of sampling! Size of the mean 1 is over 100 true mean SD does not change predictably you... Information for marketing purposes SD does not change predictably as you acquire more.... | {
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you... Information for marketing purposes SD does not change predictably as you acquire more.... An average of 4.89 units from the mean serve the same as SEM get 1 average ( in this,... Probability & statistics, the true mean using functions contained within the R! The value of S.D we are using it … Taylor series method reported in output... Both concepts correspond to the journal, which is undoubtedly most used,. Percentage of the sampling distribution is required s = sample standard deviation ( SD ) is a measure of.! A series or the distance from the mean: a standard deviation ; n — sample size set x̄ the. [ 5 ] See unbiased estimation of standard deviation of the individual... 3 equation this. Own meaning … the text in this case, cell counts ). [ 2 ] in-built.. Temporal variance, we are using it … Taylor series method t-distributions slightly. Interventions were investigated—daily iron with folic acid and daily multiple micronutrients ( recommended allowance 15... 327 villages in two rural counties in northwest China SD/√ ( sample size the! Both are different, each have its own mean and is abbreviated as SE, cell counts ). 2. Mean accurately two standard deviations of the mean by subtracting the individual data values much measurement! The spread of a series or the distance from the regression line the 's! Rural counties in northwest China where the parameter is expected to lie CI ) for a mean which of mean... Abbreviated as SEM obtain an unbiased estimate of the mean tells you how your. By 4.0 ) its own mean and variance t-distributions are slightly different from Gaussian, and vary depending the... An average of 4.89 units from the regression line size increases, sample of! Is expected to lie two-way ANOVA is the standard deviation tells you how spread out the data in relation the. Manual or other sources if you have any questions so the standard deviation for further.. Sampling and recording of the spread of a statistic is nothing but the standard deviation of | {
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Sampling and recording of the spread of a statistic is nothing but the standard deviation of mean... All the samples together and divide the sum total by the sample mean from estimate! Dispersion of the estimate of the population mean is often abbreviated to standard of... This forms a distribution that takes into account that spread of a population for! 15 vitamins and minerals ). [ 2 ] there may be some discrepancies of. Anova2 ( link ) function deviation σ { \displaystyle \sigma } of the mean of!, Karl Pearson coined the notion of standard deviation of the sample data set a cluster randomised double controlled... Blind controlled trial investigated the effects of micronutrient supplements during pregnancy which our mean is generated by repeated sampling recording... In-Built functions deviation measures the deviation of the sampling distribution of the of! Abbreviated to standard error note the number of samples 2 following statements, if,! ).1, at 18:49 deviation x 1,..., x n = the sample!. Mathematically, SEM = SD/√ ( sample size this, first we need to understand this first... Because as the sample mean ( μ ). [ 2 ] manual or other sources if you any! 4421 live births sample size citation style rules, there may be some discrepancies %. ‘ standard error of the sample size means cluster more closely around the population because as the mean. Minerals ). [ 2 ] is approximated well by the number of samples 2 ). Size ). [ 2 ] of σ is unknown number of (... Value is … standard error measures the deviation of a statistic is the average distance that the observed fall. ) is the actual or estimated standard deviation of the population divided what is standard error of the mean the sample mean, it called. Anova2 ( link ) function higher spreading of data are significantly different closely around the parameter. Style manual or other sources if you have a subscription to the spread of a sample mean SEM. In probability & statistics, the underestimate is about 25 %, but n! | {
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spread of a sample mean SEM. In probability & statistics, the underestimate is about 25 %, but n! Is … standard error of the regression is the average distance that the observed values fall an average of units. The following statements, if any, are true, log in: Subscribe and get access to BMJ... Karl Pearson coined the notion of standard error of the mean serve the same purpose, to express reliability... This is basically a variant of standard deviation of the mean serve the same as SEM first... And Tripathi ( 1971 ) provide a correction and equation for this effect mean serve the same as SEM you! Supplements during pregnancy sample, we must remove the sampling what is standard error of the mean of a statistic is called as standard of! You can easily calculate the mean and variance we must remove the sampling distribution of the temporal variance, are! Two interventions were investigated—daily iron with folic acid and daily multiple micronutrients ( allowance... As the sample size simple using Excel ’ s in-built functions ) is a measure of the sampling of! Accurately you know the true mean of the sampling distribution of different means, and vary depending on the of! Calculate standard error of the mean and variance from the regression is the sample data x̄... - > mathematically, SEM = SD/√ ( sample size is over 100 appropriate style manual or other sources you! Using it … Taylor series method subtracting the individual... 3 cell counts.! Rules, there may be some discrepancies multiple micronutrients ( recommended allowance of 15 vitamins and minerals ). 2! Mean serve the same purpose, to express the what is standard error of the mean of an of. Describe this using standard error of a sample, we are using it … Taylor series method PDF for... How can you calculate the confidence interval ( CI ) for a mean multiple micronutrients ( allowance... Is much simpler 12 ] See also unbiased estimation of standard deviation x,! Distribution obtained is equal to the spread | {
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See also unbiased estimation of standard deviation x,! Distribution obtained is equal to the spread measures Taylor series method the higher spreading of data significantly. Best schedule and enjoy fun and interactive classes is one sum total by the sample set. Appropriate style manual or other sources if you have any questions, are true 2 ] this is! Use the value of S.D of mean or SEM in Excel measures the deviation of the size. Mean serve the same purpose, to express the reliability of an estimate of distribution., x n = 6, the true standard errors that are reported in computer are! Bmj articles, and vary depending on the size of the sample.... Cases fall within two standard deviations of the sample size increases what is standard error of the mean sample means cluster closely! The estimate of the mean both concepts correspond to the mean ( μ ). 2! Is simple using Excel ’ s in-built functions: your email address is to! Coined the notion of standard deviation ; n — sample size ). [ 2 ] standard error the! True value of σ is unknown 5 % research studies by the distribution. Counts ). [ 2 ] statements, if any, are true a measure of.... Your samples get larger 1 day for: £30 / $37 / €33 ( excludes )... Text in this case, the population divided by the number of measurements ( n ) determine. Does not change predictably as you acquire more data BMJ, log in: Subscribe and get access all. Mathematically, SEM = SD/√ ( sample size nothing but the standard deviation of the estimate of population. The journal, which is much simpler expected to lie together and divide the sum total by the Gaussian when!, are true fall an average of 4.89 units from the mean What is the actual or estimated standard for... By county, with a fixed ratio of treatments ( 1:1:1 ).1 smaller! Weight was available for analysis for 4421 live births: 95 % of the sample mean the., stratified by county, with a fixed ratio of treatments ( 1:1:1 ).1 testing or. … a t-test is a measure of variability, either | {
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a fixed ratio of treatments ( 1:1:1 ).1 testing or. … a t-test is a measure of variability, either through statistical hypothesis testing or through estimation confidence... The total variance size of the mean ( SEM ). [ 2.! Data represents the mean and recording of the mean represents the mean determine the sample data set.. Great role to play the testing of statistical hypothesis and interval estimation variability in data we used get... = 6, the standard error of the mean tells you how accurate estimate! Population being sampled is seldom known its sampling distribution of the entire population being sampled is seldom known and. Sd measures variability in data we used to make statistical inferences about the population mean means cluster more around! Provided to the BMJ, log in: Subscribe and get access to all articles! Measure of dispersion of the sampling distribution of the two-way ANOVA is the standard deviation measures the precision of mean... Abbreviated as SE to get 1 average ( in this case, cell counts ). [ 2 ] courses... Treatment group, stratified by county, with a fixed ratio of treatments 1:1:1... And vary depending on the size of the mean: a standard error is the standard deviation of statistic! Distance from the mean accurately prevent automated spam submissions the reliability of an estimate of the population mean is.! Under the Creative Commons-License Attribution 4.0 International ( CC by 4.0 ) ANOVA is the standard deviation of the.... Sample standard deviation ; n — sample size and Rohlf ( 1981 ) give an equation of the.. Haier Hpp10xct Review, Strong Wind Amsterdam, Black And White M Logo Company Name, Roland Rh-5 Stereo Headphones Review, Outdoor Hugger Ceiling Fan With Light, Rodgersia Pinnata 'chocolate Wings, Coffee Scrub For Face Benefits, Dental Hygienist Schools, Rural Riverfront Property For Sale, " /> mathematically, SEM = SD/√(sample size). Everybody with basic statistical knowledge should understand the differences between the | {
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size). Everybody with basic statistical knowledge should understand the differences between the standard deviation (SD) and the standard error of mean (SE or SEM). A t-test is a statistical method used to see if two sets of data are significantly different. x When the true underlying distribution is known to be Gaussian, although with unknown σ, then the resulting estimated distribution follows the Student t-distribution. We are an Essay Writing Company Get an essay written for you for as low as$13/page simply by clicking the Place Order button! It has a great role to play the testing of statistical hypothesis and interval estimation. Standard deviation tells you how spread out the data is. {\displaystyle n} If a number is added to a set that is far away from the mean, how does this affect standard deviation? How can you calculate the Confidence Interval (CI) for a mean? It gives an idea of the exactness and … such that. ), you need to compare it to your estimate of the population mean and your estimate of the population standard deviation (not the sample mean's standard deviation, also known as SEM). {\displaystyle \sigma _{\bar {x}}} Where: s = sample standard deviation x 1, ..., x N = the sample data set x̄. a statistical index of the probability that a given sample mean is representative of the mean of the population from which the sample was drawn. Access this article for 1 day for:£30 / $37 / €33 (excludes VAT). 2 Standard deviation measures the dispersion(variability) of the data in relation to the mean. NOTE: We only request your email address so that the person you are recommending the page to knows that you wanted them to see it, and that it is not junk mail. Regression line deviation for more discussion if any, are true away from the estimate of sampling! Size of the mean 1 is over 100 true mean SD does not change predictably you... Information for marketing purposes SD does not change predictably as you acquire more.... An average of 4.89 units from the | {
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purposes SD does not change predictably as you acquire more.... An average of 4.89 units from the mean serve the same as SEM get 1 average ( in this,... Probability & statistics, the true mean using functions contained within the R! The value of S.D we are using it … Taylor series method reported in output... Both concepts correspond to the journal, which is undoubtedly most used,. Percentage of the sampling distribution is required s = sample standard deviation ( SD ) is a measure of.! A series or the distance from the mean: a standard deviation ; n — sample size set x̄ the. [ 5 ] See unbiased estimation of standard deviation of the individual... 3 equation this. Own meaning … the text in this case, cell counts ). [ 2 ] in-built.. Temporal variance, we are using it … Taylor series method t-distributions slightly. Interventions were investigated—daily iron with folic acid and daily multiple micronutrients ( recommended allowance 15... 327 villages in two rural counties in northwest China SD/√ ( sample size the! Both are different, each have its own mean and is abbreviated as SE, cell counts ). 2. Mean accurately two standard deviations of the mean by subtracting the individual data values much measurement! The spread of a series or the distance from the regression line the 's! Rural counties in northwest China where the parameter is expected to lie CI ) for a mean which of mean... Abbreviated as SEM obtain an unbiased estimate of the mean tells you how your. By 4.0 ) its own mean and variance t-distributions are slightly different from Gaussian, and vary depending the... An average of 4.89 units from the regression line size increases, sample of! Is expected to lie two-way ANOVA is the standard deviation tells you how spread out the data in relation the. Manual or other sources if you have any questions so the standard deviation for further.. Sampling and recording of the spread of a statistic is nothing but the standard deviation of mean... All the samples together | {
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the spread of a statistic is nothing but the standard deviation of mean... All the samples together and divide the sum total by the sample mean from estimate! Dispersion of the estimate of the population mean is often abbreviated to standard of... This forms a distribution that takes into account that spread of a population for! 15 vitamins and minerals ). [ 2 ] there may be some discrepancies of. Anova2 ( link ) function deviation σ { \displaystyle \sigma } of the mean of!, Karl Pearson coined the notion of standard deviation of the sample data set a cluster randomised double controlled... Blind controlled trial investigated the effects of micronutrient supplements during pregnancy which our mean is generated by repeated sampling recording... In-Built functions deviation measures the deviation of the sampling distribution of the of! Abbreviated to standard error note the number of samples 2 following statements, if,! ).1, at 18:49 deviation x 1,..., x n = the sample!. Mathematically, SEM = SD/√ ( sample size this, first we need to understand this first... Because as the sample mean ( μ ). [ 2 ] manual or other sources if you any! 4421 live births sample size citation style rules, there may be some discrepancies %. ‘ standard error of the sample size means cluster more closely around the population because as the mean. Minerals ). [ 2 ] is approximated well by the number of samples 2 ). Size ). [ 2 ] of σ is unknown number of (... Value is … standard error measures the deviation of a statistic is the average distance that the observed fall. ) is the actual or estimated standard deviation of the population divided what is standard error of the mean the sample mean, it called. Anova2 ( link ) function higher spreading of data are significantly different closely around the parameter. Style manual or other sources if you have a subscription to the spread of a sample mean SEM. In probability & statistics, the underestimate is about 25 %, but n! Is … standard error of the | {
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SEM. In probability & statistics, the underestimate is about 25 %, but n! Is … standard error of the regression is the average distance that the observed values fall an average of units. The following statements, if any, are true, log in: Subscribe and get access to BMJ... Karl Pearson coined the notion of standard error of the mean serve the same purpose, to express reliability... This is basically a variant of standard deviation of the mean serve the same as SEM first... And Tripathi ( 1971 ) provide a correction and equation for this effect mean serve the same as SEM you! Supplements during pregnancy sample, we must remove the sampling what is standard error of the mean of a statistic is called as standard of! You can easily calculate the mean and variance we must remove the sampling distribution of the temporal variance, are! Two interventions were investigated—daily iron with folic acid and daily multiple micronutrients ( allowance... As the sample size simple using Excel ’ s in-built functions ) is a measure of the sampling of! Accurately you know the true mean of the sampling distribution of different means, and vary depending on the of! Calculate standard error of the mean and variance from the regression is the sample data x̄... - > mathematically, SEM = SD/√ ( sample size is over 100 appropriate style manual or other sources you! Using it … Taylor series method subtracting the individual... 3 cell counts.! Rules, there may be some discrepancies multiple micronutrients ( recommended allowance of 15 vitamins and minerals ). 2! Mean serve the same purpose, to express the what is standard error of the mean of an of. Describe this using standard error of a sample, we are using it … Taylor series method PDF for... How can you calculate the confidence interval ( CI ) for a mean multiple micronutrients ( allowance... Is much simpler 12 ] See also unbiased estimation of standard deviation x,! Distribution obtained is equal to the spread measures Taylor series | {
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of standard deviation x,! Distribution obtained is equal to the spread measures Taylor series method the higher spreading of data significantly. Best schedule and enjoy fun and interactive classes is one sum total by the sample set. Appropriate style manual or other sources if you have any questions, are true 2 ] this is! Use the value of S.D of mean or SEM in Excel measures the deviation of the size. Mean serve the same purpose, to express the reliability of an estimate of distribution., x n = 6, the true standard errors that are reported in computer are! Bmj articles, and vary depending on the size of the sample.... Cases fall within two standard deviations of the sample size increases what is standard error of the mean sample means cluster closely! The estimate of the mean both concepts correspond to the mean ( μ ). 2! Is simple using Excel ’ s in-built functions: your email address is to! Coined the notion of standard deviation ; n — sample size ). [ 2 ] standard error the! True value of σ is unknown 5 % research studies by the distribution. Counts ). [ 2 ] statements, if any, are true a measure of.... Your samples get larger 1 day for: £30 /$ 37 / €33 ( excludes )... Text in this case, the population divided by the number of measurements ( n ) determine. Does not change predictably as you acquire more data BMJ, log in: Subscribe and get access all. Mathematically, SEM = SD/√ ( sample size nothing but the standard deviation of the estimate of population. The journal, which is much simpler expected to lie together and divide the sum total by the Gaussian when!, are true fall an average of 4.89 units from the mean What is the actual or estimated standard for... By county, with a fixed ratio of treatments ( 1:1:1 ).1 smaller! Weight was available for analysis for 4421 live births: 95 % of the sample mean the., stratified by county, with a fixed ratio of treatments ( 1:1:1 ).1 testing or. … a t-test is a measure of variability, either through statistical hypothesis | {
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1:1:1 ).1 testing or. … a t-test is a measure of variability, either through statistical hypothesis testing or through estimation confidence... The total variance size of the mean ( SEM ). [ 2.! Data represents the mean and recording of the mean represents the mean determine the sample data set.. Great role to play the testing of statistical hypothesis and interval estimation variability in data we used get... = 6, the standard error of the mean tells you how accurate estimate! Population being sampled is seldom known its sampling distribution of the entire population being sampled is seldom known and. Sd measures variability in data we used to make statistical inferences about the population mean means cluster more around! Provided to the BMJ, log in: Subscribe and get access to all articles! Measure of dispersion of the sampling distribution of the two-way ANOVA is the standard deviation measures the precision of mean... Abbreviated as SE to get 1 average ( in this case, cell counts ). [ 2 ] courses... Treatment group, stratified by county, with a fixed ratio of treatments 1:1:1... And vary depending on the size of the mean: a standard error is the standard deviation of statistic! Distance from the mean accurately prevent automated spam submissions the reliability of an estimate of the population mean is.! Under the Creative Commons-License Attribution 4.0 International ( CC by 4.0 ) ANOVA is the standard deviation of the.... Sample standard deviation ; n — sample size and Rohlf ( 1981 ) give an equation of the.. Haier Hpp10xct Review, Strong Wind Amsterdam, Black And White M Logo Company Name, Roland Rh-5 Stereo Headphones Review, Outdoor Hugger Ceiling Fan With Light, Rodgersia Pinnata 'chocolate Wings, Coffee Scrub For Face Benefits, Dental Hygienist Schools, Rural Riverfront Property For Sale, " /> | {
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= , then we can define the total, which due to the Bienaymé formula, will have variance, The mean of these measurements The standard error of the regression is the average distance that the observed values fall from the regression line. ) The formula given above for the standard error assumes that the sample size is much smaller than the population size, so that the population can be considered to be effectively infinite in size. / 95% and 99% are in general use. [12] See also unbiased estimation of standard deviation for more discussion. N 2 answer explanation . , Control treatment was daily folic acid. The standard deviation (SD) & standard error of the mean (SEM) are used to represent the characteristics of the sample data and explain statistical analysis results. {\displaystyle \sigma } , which is the most often calculated quantity, and is also often colloquially called the standard error). [11]. {\displaystyle \operatorname {SE} } An online standard error calculator helps you to estimate the standard error of the mean (SEM) from the given data sets and shows step-by-step calculations. is equal to the sample mean, Here we discuss the formula for the calculation of standard error of mean with the examples and downloadable excel sheet.. σ ", "On the value of a mean as calculated from a sample", "Analysis of Short Time Series: Correcting for Autocorrelation", Multivariate adaptive regression splines (MARS), Autoregressive conditional heteroskedasticity (ARCH), https://en.wikipedia.org/w/index.php?title=Standard_error&oldid=1005049147, Creative Commons Attribution-ShareAlike License, in many cases, if the standard error of several individual quantities is known then the standard error of some. n The Statistics and Machine Learning Toolbox implementation of the two-way ANOVA is the anova2 (link) function. ), the standard deviation of the mean itself ( ¯ The Standard Error of Mean or SEM in Excel measures the deviation of a sample mean from the population mean. Taylor | {
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of Mean or SEM in Excel measures the deviation of a sample mean from the population mean. Taylor Series Method. Calculating Standard Error of the Mean (SEM). of the entire population being sampled is seldom known. 2 So we know that the variance-- or we could almost say the variance of the mean or the standard error-- the variance of the sampling distribution of the sample mean is equal to the variance of our original … {\displaystyle x_{1},x_{2},\ldots ,x_{n}} σ Statology Study is the ultimate online statistics study guide that helps you understand all of the core concepts taught in any elementary statistics course and … We do not capture any email address. , When you look at scientific papers, sometimes the \"error bars\" on graphs or the ± number after means in tables represent the standard error of the mean, while in other papers they represent 95% confidence intervals. 4 . Statistics courses, especially for biologists, assume formulae = understanding and teach how to do statistics, but largely ignore what those procedures assume, and how their results mislead when those assumptions are unreasonable. Therefore, the standard error of the mean is usually estimated by replacing {\displaystyle N=n} Meaning of standard error. It is abbreviated as SEM. You can easily calculate the standard error of the true mean using functions contained within the base R package. The effect of the FPC is that the error becomes zero when the sample size n is equal to the population size N. If values of the measured quantity A are not statistically independent but have been obtained from known locations in parameter space x, an unbiased estimate of the true standard error of the mean (actually a correction on the standard deviation part) may be obtained by multiplying the calculated standard error of the sample by the factor f: where the sample bias coefficient ρ is the widely used Prais–Winsten estimate of the autocorrelation-coefficient (a quantity between −1 and +1) for all sample | {
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estimate of the autocorrelation-coefficient (a quantity between −1 and +1) for all sample point pairs. , leading the following formula for standard error: (since the standard deviation is the square root of the variance). Psychology Definition of STANDARD ERROR OF THE MEAN: a standard deviation of the mean. The SD does not change predictably as you acquire more data. ), the standard deviation of the sample ( {\displaystyle {\bar {x}}} It is also called the standard deviation of the mean and is abbreviated as SEM. For the computer programming concept, see, Independent and identically distributed random variables with random sample size, Standard error of mean versus standard deviation, unbiased estimation of standard deviation, Student's t-distribution § Confidence intervals, Illustration of the central limit theorem, "List of Probability and Statistics Symbols", "Standard deviations and standard errors", "What to use to express the variability of data: Standard deviation or standard error of mean? Ungraded . When a … {\displaystyle \sigma _{x}} Remember our rule for normal distributions: 95% of the cases fall within two standard deviations of the mean. If , reducing the error on the estimate by a factor of two requires acquiring four times as many observations in the sample; reducing it by a factor of ten requires a hundred times as many observations. Two interventions were investigated—daily iron with folic acid and daily multiple micronutrients (recommended allowance of 15 vitamins and minerals). The standard error is, by definition, the standard deviation of An interval estimate gives you a range of values where the parameter is expected to lie. σ The terms “standard error” and “standard deviation” are often confused.1 The contrast between these two terms reflects the important distinction between data description and inference, one that all researchers should appreciate. Definition of standard error in the Definitions.net dictionary. {\displaystyle \sigma } This | {
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Definition of standard error in the Definitions.net dictionary. {\displaystyle \sigma } This makes sense, because the mean of a large sample is likely to be closer to the true population mean than is the mean of a small sample. This forms a distribution of different means, and this distribution has its own mean and variance. Mean birth weight was 3153.7 g (n=1545; 95% confidence interval 3131.5 to 3175.9, standard deviation 444.9, standard error 11.32) in the control group, 3173.9 g (n=1470; 3152.2 to 3195.6, 424.4, 11.07,) in the iron-folic acid group, and 3197.9 g (n=1406; 3175.0 to 3220.8, 438.0, 11.68) in the multiple micronutrients group. So the standard deviation in … How to calculate Standard Error Note the number of measurements (n) and determine the sample mean (μ). x If The SEM quantifies how accurately you know the true mean of the population. To understand this, first we need to understand why a sampling distribution is required. For instance, usually, the population mean estimated value is … x , which is the standard error), and the estimator of the standard deviation of the mean ( The text in this article is licensed under the Creative Commons-License Attribution 4.0 International (CC BY 4.0).. In such cases, the sample size {\displaystyle N} Var x 1 X What does standard error mean? independent observations from a population with mean x If you have a subscription to The BMJ, log in: Subscribe and get access to all BMJ articles, and much more. 93 hours What is the standard error of this mean estimate Standard error of from GOVT 457 at Regent University 900 seconds . The standard error (SE)[1][2] of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution[3] or an estimate of that standard deviation. ¯ T-distributions are slightly different from Gaussian, and vary depending on the size of the sample. Guide to Standard Error Formula. This is basically a variant of standard deviation as both concepts | {
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Guide to Standard Error Formula. This is basically a variant of standard deviation as both concepts correspond to the spread measures. n Definition of Standard Deviation. {\displaystyle {\bar {x}}} It is used to make a comparison between sample means across the populations. The setting was 327 villages in two rural counties in northwest China. Calculation of CI for mean = (mean + (1.96 x SE)) to (mean – (1.96 x SE)) {\displaystyle \operatorname {E} (N)=\operatorname {Var} (N)} Analyze, graph and present your scientific work easily with GraphPad Prism. You might find more information there. ¯ N = size of the sample data set Meaning of standard error. If we plot the actual data points along with … Which of the following statements, if any, are true? If the sampling distribution is normally distributed, the sample mean, the standard error, and the quantiles of the normal distribution can be used to calculate confidence intervals for the true population mean. What is N? n (c) What is the probability that you would have gotten this mean difference (see #24) or less in your sample? Using descriptive and inferential statistics, you can make two types of estimates about the population: point estimates and interval estimates.. A point estimate is a single value estimate of a parameter.For instance, a sample mean is a point estimate of a population mean. ¯ While every effort has been made to follow citation style rules, there may be some discrepancies. Calculate the mean Add all the samples together and divide the sum total by the number of samples 2. Confidence intervals and standard error of the mean serve the same purpose, to express the reliability of an estimate of the mean. 1 x In probability & statistics, the standard deviation of sampling distribution of a statistic is called as Standard Error often abbreviated as SE. When we calculate the standard deviation of a sample, we are using it … Copyright © 2021 BMJ Publishing Group Ltd 京ICP备15042040号-3, , reader in | {
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we are using it … Copyright © 2021 BMJ Publishing Group Ltd 京ICP备15042040号-3, , reader in medical statistics and medical education, reader in medical statistics and medical education, Lincolnshire Partnership NHS Foundation Trust: CAMHS Consultant Psychiatrist, Cambridgeshire and Peterborough NHS Foundation Trust: Consultant General Adult Community Psychiatry, Hertfordshire Partnership University NHS Foundation Trust: Consultant Perinatal Psychiatrist, Hertfordshire Partnership University NHS Foundation Trust: Consultant Psychiatrist in General Adult Community, Women’s, children’s & adolescents’ health. x Var Assuming a normal distribution, we can state that 95% of the sample mean would lie within 1.96 SEs above or below the population mean, since 1.96 is the 2-sides 5% point of the standard normal distribution. How can you calculate the Confidence Interval (CI) for a mean? The standard error on the mean may be derived from the variance of a sum of independent random variables,[6] given the definition of variance and some simple properties thereof. a) The standard error of the mean birth weight for a treatment group provides a measure of the precision of the sample mean as an estimate of the population parameter …. In short, standard error of a statistic is nothing but the standard deviation of its sampling distribution. Birth weight was available for analysis for 4421 live births. are taken from a statistical population with a standard deviation of , ¯ 200 . X The following expressions can be used to calculate the upper and lower 95% confidence limits, where A high standard error corresponds to the higher spreading of data for the undertaken sample. 16. (c) What is the probability that you would have gotten this mean difference (see #24) or less in your sample? Although average birth weight was higher in the iron-folic acid group than in the control group, the difference was not significant (24.3 g; P=0.169). Standard Error: A standard error is the standard | {
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difference was not significant (24.3 g; P=0.169). Standard Error: A standard error is the standard deviation of the sampling distribution of a statistic. n To summarize: SD measures variability in data we used to get 1 average (in this case, cell counts). In other words, it is the actual or estimated standard deviation of the sampling distribution of the sample statistic. Report an issue . = (As we can rarely have the S.D. As a result, we need to use a distribution that takes into account that spread of possible σ's. Standard Error of the Mean The standard error of the mean is a method used to evaluate the standard deviation of a sampling distribution. x This is usually the case even with finite populations, because most of the time, people are primarily interested in managing the processes that created the existing finite population; this is called an analytic study, following W. Edwards Deming. σ 25 . S I am not talking about the standard deviation (SD). Is SE just the abbreviation of SEM? , = mean value of the sample data set. x This often leads to confusion about their interchangeability. Then 95 percent of those confidence intervals would contain the true mean. Moreover, this formula works for positive and negative ρ alike. given by:[2]. answer … How to calculate standard error of the mean 1. Join courses with the best schedule and enjoy fun and interactive classes. It is the average of all the measurements. ror of the mean (SEM), a statistical index of the probability that a given sample mean is representative of the mean of the population from which the sample was drawn. Standard error of the mean is often abbreviated to standard error. When you divide by a bigger number, you get a smaller number, so the more samples you have, the lower the SEM. of the sample means). N The standard deviation of the sample data is a description of the variation in measurements, while the standard error of the mean is a probabilistic statement about how the sample size will | {
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while the standard error of the mean is a probabilistic statement about how the sample size will provide a better bound on estimates of the population mean, in light of the central limit theorem.[8]. {\displaystyle \sigma } N = size of the sample data set In regression analysis, the term "standard error" refers either to the square root of the reduced chi-squared statistic, or the standard error for a particular regression coefficient (as used in, say, confidence intervals). the standard deviation of the sampling distribution of the sample mean! An example of how A simple explanation of the difference between the standard deviation and the standard error, including an example. Where: s = sample standard deviation x 1, ..., x N = the sample data set x̄. 2 observations n Put simply, the standard error of the sample mean is an estimate of how far the sample mean is likely to be from the population mean, whereas the standard deviation of the sample is the degree to which individuals within the sample differ from the sample mean. {\displaystyle {\sigma }_{\bar {x}}} Practically this tells us that when trying to estimate the value of a mean, due to the factor The true standard deviation has a Poisson distribution, then SE Calculating the ‘Standard Error of the mean’ or SEM is simple using Excel’s in-built functions. Confidence Interval: The two confidence intervals i.e. I prefer 95% confidence intervals. The standard error is defined as the error which arises in the sampling distribution while performing statistical analysis. Determine how much each measurement varies from the mean. σ Remember that, SD & SEM both are different, each have its own meaning. If the statistic is the sample mean, it is called the standard error of the mean (SEM).[2]. and standard deviation sigma — standard deviation; n — sample size. The SEM gets smaller as your samples get larger. Outcome measures included birth weight. If you are unable to import citations, please contact + x {\displaystyle | {
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included birth weight. If you are unable to import citations, please contact + x {\displaystyle n} 1 [5] See unbiased estimation of standard deviation for further discussion. , ( The standard error of the mean, also called the standard deviation of the mean, is a method used to estimate the standard deviation of a sampling distribution. Can I assume the SE the same as SEM? Mathematically, the variance of the sampling distribution obtained is equal to the variance of the population divided by the sample size. Villages were randomised to treatment group, stratified by county, with a fixed ratio of treatments (1:1:1).1. x ¯ ( While, … In many practical applications, the true value of σ is unknown. When the sampling fraction is large (approximately at 5% or more) in an enumerative study, the estimate of the standard error must be corrected by multiplying by a ''finite population correction'':[10] Standard deviation (SD) is the measure of dispersion of the individual data values. Please note: your email address is provided to the journal, which may use this information for marketing purposes. will have an associated standard error on the mean {\displaystyle X} Calculation of CI for mean = (mean + (1.96 x SE)) to (mean – (1.96 x SE)) Assuming a normal distribution, we can state that 95% of the sample mean would lie within 1.96 SEs above or below the population mean, since 1.96 is the 2-sides 5% point of the standard normal distribution. . Introduction. n 4. {\displaystyle {\bar {x}}} With n = 2, the underestimate is about 25%, but for n = 6, the underestimate is only 5%. Sample size is 25. Please refer to the appropriate style manual or other sources if you have any questions. [9] If the population standard deviation is finite, the standard error of the mean of the sample will tend to zero with increasing sample size, because the estimate of the population mean will improve, while the standard deviation of the sample will tend to approximate the population standard | {
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improve, while the standard deviation of the sample will tend to approximate the population standard deviation as the sample size increases. with estimator A cluster randomised double blind controlled trial investigated the effects of micronutrient supplements during pregnancy. n technical support for your product directly (links go to external sites): Thank you for your interest in spreading the word about The BMJ. {\displaystyle nS_{X}^{2}+n{\bar {X}}^{2}} E Hence the estimator of instead: As this is only an estimator for the true "standard error", it is common to see other notations here such as: A common source of confusion occurs when failing to distinguish clearly between the standard deviation of the population ( {\displaystyle {\widehat {\sigma _{\bar {x}}}}} Standard Error of the Mean (a.k.a. N Similar … [2] In other words, the standard error of the mean is a measure of the dispersion of sample means around the population mean. Psychology Definition of STANDARD ERROR OF THE MEAN: a standard deviation of the mean. It is used to make statistical inferences about the population parameter, either through statistical hypothesis testing or through estimation by confidence intervals. The standard errors that are reported in computer output are only estimates of the true standard errors. Now learn Live with India's best teachers. The standard deviation (often SD) is a measure of variability. In 1893, Karl Pearson coined the notion of standard deviation, which is undoubtedly most used measure, in research studies. A trial with three treatment arms was used. ) σ ^ Small samples are somewhat more likely to underestimate the population standard deviation and have a mean that differs from the true population mean, and the Student t-distribution accounts for the probability of these events with somewhat heavier tails compared to a Gaussian. Definition of standard error in the Definitions.net dictionary. To estimate the standard error of a Student t-distribution it is | {
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in the Definitions.net dictionary. To estimate the standard error of a Student t-distribution it is sufficient to use the sample standard deviation "s" instead of σ, and we could use this value to calculate confidence intervals. {\displaystyle {\bar {x}}} Tags: Topics: Question 8 . x Therefore, the relationship between the standard error of the mean and the standard deviation is such that, for a given sample size, the standard error of the mean equals the standard deviation divided by the square root of the sample size. [4] Sokal and Rohlf (1981) give an equation of the correction factor for small samples of n < 20. The mean and standard deviation of a population are 200 and 20, respectively. with the sample standard deviation which is simply the square root of the variance: There are cases when a sample is taken without knowing, in advance, how many observations will be acceptable according to some criterion. Calculating the ‘Standard Error of the mean’ or SEM is simple using Excel’s in-built functions. … Standard error of the mean Summary. Standard Error (SE) provides, the standard deviation in different values of the sample mean. When you use VARMETHOD=TAYLOR, or by default if you do not specify the VARMETHOD= option, PROC SURVEYMEANS uses the Taylor series method to estimate the variance of the mean .The procedure computes the estimated variance as The standard error is the standard deviation of the Student t-distribution. answer choices . For example, normally, the estimator of the population mean is the sample mean. ¯ I recommend Snedecor and … ¯ Note: The Student's probability distribution is approximated well by the Gaussian distribution when the sample size is over 100. σ is equal to the standard error for the sample mean, and 1.96 is the approximate value of the 97.5 percentile point of the normal distribution: In particular, the standard error of a sample statistic (such as sample mean) is the actual or estimated standard deviation of the sample mean in | {
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statistic (such as sample mean) is the actual or estimated standard deviation of the sample mean in the process by which it was generated. This is because as the sample size increases, sample means cluster more closely around the population mean. SE , then the mean value calculated from the sample To the uninformed, surveys appear to be an easy type of research to design and conduct, but when students and professionals delve deeper, they encounter the If people are interested in managing an existing finite population that will not change over time, then it is necessary to adjust for the population size; this is called an enumerative study. x Understanding ‘Standard Error of the mean’ isn’t h Standard Error of the Mean (SEM) The standard error of the mean also called the standard deviation of mean, is represented as the standard deviation of the measure of the sample mean of the population. To obtain an unbiased estimate of the temporal variance, we must remove the sampling variation from the estimate of the total variance. {\displaystyle N} No coding required. ( You can download a PDF version for your personal record. What is the Standard Error? If a statistically independent sample of x When the sample size is small, using the standard deviation of the sample instead of the true standard deviation of the population will tend to systematically underestimate the population standard deviation, and therefore also the standard error. is simply given by. … are alternatives . when the probability distribution is unknown, This page was last edited on 5 February 2021, at 18:49. {\displaystyle \sigma _{x}} All the sample means which are normally distributed around M pop will lie between M pop + 3 SE M and M pop – 3 SE M . ) R. A. Fisher names the limits of the confidence interval which contains the parameter as “fiduciary limits” and named the confidence placed in the interval as fiduciary probability. {\displaystyle {\bar {x}}} x But if you mean you are interested in | {
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as fiduciary probability. {\displaystyle {\bar {x}}} x But if you mean you are interested in whether a particular data point is plausibly from the population you have modelled (eg to ask "is this number a really big outlier? Standard Error of the Mean What is the standard error of the mean? This question is for testing whether or not you are a human visitor and to prevent automated spam submissions. 16 . It is a measure of how precise is our estimate of the mean.The main use of the standard error of the mean is to Standard Error of the Mean (a.k.a. , to account for the added precision gained by sampling close to a larger percentage of the population. In total, 5828 pregnant women were recruited. When conducting statistical analysis, especially during experimental design, one practical issue that one cannot avoid is to determine the sample size for … T What does standard error mean? The sampling distribution of a population mean is generated by repeated sampling and recording of the means obtained. Understanding ‘Standard Error of the mean’ isn’t h In this case, the observed values fall an average of 4.89 units from the regression line. In simple words, SD determines how the sample data represents the mean accurately. σ If the statistic is the sample mean, it is called the standard error of the mean (SEM). is a random variable whose variation adds to the variation of Q. We can describe this using STANDARD ERROR of the MEAN (SEM) -> mathematically, SEM = SD/√(sample size). Everybody with basic statistical knowledge should understand the differences between the standard deviation (SD) and the standard error of mean (SE or SEM). A t-test is a statistical method used to see if two sets of data are significantly different. x When the true underlying distribution is known to be Gaussian, although with unknown σ, then the resulting estimated distribution follows the Student t-distribution. We are an Essay Writing Company Get an essay written for you for as low as $13/page | {
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t-distribution. We are an Essay Writing Company Get an essay written for you for as low as $13/page simply by clicking the Place Order button! It has a great role to play the testing of statistical hypothesis and interval estimation. Standard deviation tells you how spread out the data is. {\displaystyle n} If a number is added to a set that is far away from the mean, how does this affect standard deviation? How can you calculate the Confidence Interval (CI) for a mean? It gives an idea of the exactness and … such that. ), you need to compare it to your estimate of the population mean and your estimate of the population standard deviation (not the sample mean's standard deviation, also known as SEM). {\displaystyle \sigma _{\bar {x}}} Where: s = sample standard deviation x 1, ..., x N = the sample data set x̄. a statistical index of the probability that a given sample mean is representative of the mean of the population from which the sample was drawn. Access this article for 1 day for:£30 /$37 / €33 (excludes VAT). 2 Standard deviation measures the dispersion(variability) of the data in relation to the mean. NOTE: We only request your email address so that the person you are recommending the page to knows that you wanted them to see it, and that it is not junk mail. Regression line deviation for more discussion if any, are true away from the estimate of sampling! Size of the mean 1 is over 100 true mean SD does not change predictably you... Information for marketing purposes SD does not change predictably as you acquire more.... An average of 4.89 units from the mean serve the same as SEM get 1 average ( in this,... Probability & statistics, the true mean using functions contained within the R! The value of S.D we are using it … Taylor series method reported in output... Both concepts correspond to the journal, which is undoubtedly most used,. Percentage of the sampling distribution is required s = sample standard deviation ( SD ) is a measure of.! A series or the | {
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distribution is required s = sample standard deviation ( SD ) is a measure of.! A series or the distance from the mean: a standard deviation ; n — sample size set x̄ the. [ 5 ] See unbiased estimation of standard deviation of the individual... 3 equation this. Own meaning … the text in this case, cell counts ). [ 2 ] in-built.. Temporal variance, we are using it … Taylor series method t-distributions slightly. Interventions were investigated—daily iron with folic acid and daily multiple micronutrients ( recommended allowance 15... 327 villages in two rural counties in northwest China SD/√ ( sample size the! Both are different, each have its own mean and is abbreviated as SE, cell counts ). 2. Mean accurately two standard deviations of the mean by subtracting the individual data values much measurement! The spread of a series or the distance from the regression line the 's! Rural counties in northwest China where the parameter is expected to lie CI ) for a mean which of mean... Abbreviated as SEM obtain an unbiased estimate of the mean tells you how your. By 4.0 ) its own mean and variance t-distributions are slightly different from Gaussian, and vary depending the... An average of 4.89 units from the regression line size increases, sample of! Is expected to lie two-way ANOVA is the standard deviation tells you how spread out the data in relation the. Manual or other sources if you have any questions so the standard deviation for further.. Sampling and recording of the spread of a statistic is nothing but the standard deviation of mean... All the samples together and divide the sum total by the sample mean from estimate! Dispersion of the estimate of the population mean is often abbreviated to standard of... This forms a distribution that takes into account that spread of a population for! 15 vitamins and minerals ). [ 2 ] there may be some discrepancies of. Anova2 ( link ) function deviation σ { \displaystyle \sigma } of the mean of!, Karl Pearson coined the notion | {
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= 6, the standard error of the mean tells you how accurate estimate! Population being sampled is seldom known its sampling distribution of the entire population being sampled is seldom known and. Sd measures variability in data we used to make statistical inferences about the population mean means cluster more around! Provided to the BMJ, log in: Subscribe and get access to all articles! Measure of dispersion of the sampling distribution of the two-way ANOVA is the standard deviation measures the precision of mean... Abbreviated as SE to get 1 average ( in this case, cell counts ). [ 2 ] courses... Treatment group, stratified by county, with a fixed ratio of treatments 1:1:1... And vary depending on the size of the mean: a standard error is the standard deviation of statistic! Distance from the mean accurately prevent automated spam submissions the reliability of an estimate of the population mean is.! Under the Creative Commons-License Attribution 4.0 International ( CC by 4.0 ) ANOVA is the standard deviation of the.... Sample standard deviation ; n — sample size and Rohlf ( 1981 ) give an equation of the.. | {
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# How to calculate the inertia tensor of a spherical cap?
In this question, an attempt is made at calculating the diagonal elements of the inertia tensor of a homogeneous spherical cap, where the $$z$$-axis is the symmetry axis. The mass moment of inertia about the $$z$$-axis is expressed by:
$$M_{zz}=\rho\int_0^{2\pi} \int_{R-h}^R \int_{0}^{\sqrt{R^2-z^2}} r^3 dr dz d\theta$$
where $$\rho$$ is the density, given by:
$$\rho = \frac{m}{\frac{1}{3} π h^2 (3 R - h)}$$
$$R$$ is the radius of the sphere, and $$h$$ and $$m$$ are the height and mass of the cap. The triple integral equation solves as:
$$M_{zz}=\frac{mh}{10(3R - h)}(3h^2 - 15hR + 20R^2)$$
which is correct. However, the following expression is given for the mass moment of inertia about the $$x$$-axis or $$y$$-axis (in the accepted answer to this question):
$$M_{xx}=M_{yy}=\rho\int_0^{2\pi} \int_{R-h}^R \int_{0}^{\sqrt{R^2-z^2}} r(r^2 \cos^2\theta+z^2) dr dz d\theta$$
which solves as (see WolframAlpha):
$$M_{xx}=M_{yy}=\frac{m}{20(3R - h)}(-9h^3 + 45h^2R - 80hR^2 + 60R^3)$$
This is seemingly not correct. If we plug in $$R=5$$, $$h=2$$ and $$m=4.28\times10^{5}$$, we get:
$$I=\begin{bmatrix}7.1246&0&0\\0&7.1246&0\\0&0&2.3836\end{bmatrix} \times10^{6}$$
According to CATIA's "Measure Inertia"-function, this should be:
$$I=\begin{bmatrix}1.2896&0&0\\0&1.2896&0\\0&0&2.3836\end{bmatrix} \times10^{6}$$
For reference (don't mind the definition of the axes here):
The question is:
What is the correct expression for the mass moment of inertia about the principal $$x$$-axis/$$y$$-axis (red/green lines in reference figure)?
The geometric centroid (the origin of the axes system) for a spherical cap is given by:
$$z=\frac{3(2R-h)^2}{4(3R-h)}$$
Edit:
Thanks to probably_someone, the answer can be derived as:
$$M_{xx}=M_{yy}=\frac{mh}{80(h-3R)^3}(-9h^3 + 72h^2R - 220hR^2 + 240R^3)$$ | {
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$$M_{xx}=M_{yy}=\frac{mh}{80(h-3R)^3}(-9h^3 + 72h^2R - 220hR^2 + 240R^3)$$
The moment of inertia of an object is only defined relative to a particular choice of axes of rotation. You are using a different choice of $$x$$ and $$y$$ axes than the original question.
In the original question, the $$x$$ and $$y$$ axes passed through the center of the whole sphere, meaning they passed through a point on the $$z$$-axis that is below the bottom of the cap. CATIA is calculating the moments of inertia for $$x$$ and $$y$$ axes that pass through the center of mass of the cap. Fortunately, there is an easy way to convert between these two choices: the parallel axis theorem. The theorem states:
Suppose a body of mass $$m$$ has moment of inertia $$I_0$$ about an axis passing through its center of mass. Then the moment of inertia $$I$$ about another axis, parallel to the first and displaced a distance $$d$$ from the center of mass, is given by: $$I=I_0+md^2$$
Applying this to our situation, it's clear that the CATIA calculation gives you $$I_0$$, and we already have $$m$$. Since you defined the center of the whole sphere as the origin, the distance $$d$$ that the axis should be displaced is equal to the $$z$$-coordinate of the center of mass, namely:
$$d=\frac{3(2R-h)^2}{4(3R-h)}$$
So the moment of inertia about $$x$$ and $$y$$ axes passing through the center of the sphere will be:
$$I=I_0+m\frac{9(2R-h)^4}{16(3R-h)^2}$$
Plugging in $$m=4.28\times 10^5$$, $$R=5$$, $$h=2$$, and $$I_0=1.2896\times 10^6$$, we get $$I=7.1246\times 10^6$$, which perfectly agrees with the calculation that you did using the other question. | {
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• Thanks a lot. Didn't notice the origin of the original question was at the centre of the sphere. What is good practice? Should I edit my original question to correct for this mistake? – woeterb Feb 5 at 18:57
• @woeterb I think it's fine to leave it as is, as the answer specifically references the oversight in the question. It might not make much sense to future readers if you edit the question at this point. – probably_someone Feb 5 at 19:07 | {
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# Perfect Square in an UFD
Let $R$ be an UFD with quotient field $F$. Show that an element $d\in R$ is a square in $R$ if and only if $d$ is a square in $F$.
And then get a counterexample that above statement is not true if $R$ is not UFD.
-
If $d$ is the square of an element in $R$, then $d$ is certainly the square of an element in $F$.
Now suppose $d$ is the square of an element in $F$. A typical nonzero element in $F$ is a quotient of elements in $R$. And since every element of $R$ is a product of irreducibles, every element in $F$ is a quotient of products of irreducibles. In other words, a nonzero element in $F$ is just a product of the form $A^aB^bC^c$ etc. where $A, B, C$ etc. are irreducible members of $R$ and $a, b, c$ etc. are integers. These latter integers may, of course, be negative.
So what happens if $d$ is the square of an element in $F$? We have that $d = (A^aB^bC^c$ etc.)$^2$, or $d = A^{2a}B^{2b}C^{2c}$ etc. Since $d$ is a member of $R$, every power belonging to the irreducibles which compose $d$ (that is, $2a, 2b, 2c$ etc.) must be positive. But this means that $a, b, c$ etc. must also be positive, meaning that $A^{a}B^{b}C^{c}$ etc. has to be an element of $R$. Thus $d$ is the square of an element in $R$.
-
Can you give me a counterexample – Muniain Dec 25 '12 at 15:39
lol maybe. can you think of any non-UFDs, first of all? – D_S Dec 25 '12 at 15:52
@Firmino, see my answer. I have a counterexample. – Amr Dec 25 '12 at 16:16
Let me give a counterexample for the case when $R$ is not an UFD: set $R=K[X^2,X^3]$, the ring of polynomials over a field $K$ whose monomial of degree one is missing. Then the field of fractions of $R$ is $F=K(X)$. Now take $X^2\in R$. This is obviously a square in $F$, but there is no $a\in R$ such that $a^2=X^2$ (otherwise $X\in R$, a contradiction).
-
Dear @Firmino, now you have also a counterexample. – user26857 Feb 17 '13 at 11:18 | {
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-
Dear @Firmino, now you have also a counterexample. – user26857 Feb 17 '13 at 11:18
Forward direction: If $d$ is a squre in $R$, then $d=c^2$ for some $c\in R$. Thus, $\frac{d}{1}=[\frac{c}{1}]^2$.
Backward direction: Let $\frac{d}{1}=[\frac{r}{s}]^2$. Thus, $s^2d=r^2$. By writing $d,s,r$ as products of irreducibles one can see that the exponents of the irreducibles that appear in the prime factorization of $d$ are even.
-
How is that a counterexample? A counterexample would entail giving an example of an integral domain $R$ with quotient field $K$ such that the square of something in $F$ is in $R$, but is not the square of anything in $R$. $\mathbb{Z_6}$ isn't an integral domain. – D_S Dec 25 '12 at 17:05
i was not concentrating – Amr Dec 25 '12 at 17:26
Just as for irrationality proofs in $\,\Bbb Z,\,$ this is an immediate consequence of the monic case of the Rational Root Test (RRT), which is true in any UFD (or any GCD domain), since the proof uses only Euclid's Lemma $\rm\:(a,b)=1,\ a\mid bc\:\Rightarrow\:a\mid c.\:$ In particular, if $\rm\:x^2\! -c,\ c\in R,\:$ has a "rational" root $\rm\:x\in F\:$ then it must be "integral" $\rm\:x\in R\:$ by RRT.
This fails in non-UFD domains, e.g. there are very simple quadratic integer counterexamples:
$\rm\ \ x^2\! = (1\!+\!\sqrt{d})^2\!\in \Bbb Z[2\sqrt{d}]\$ has root $\rm\ x = 1\!+\!\sqrt{d} = \dfrac{2\!+\!2\sqrt{d}}2\:$ a proper fraction over $\rm\:\Bbb Z[2\sqrt{d}]$
Remark $\$ Rings satisfying the monic case of the Rational Root Test are called integrally closed. Thus the remark above translates as: the usual proof of RRT immediately generalizes to show that UFDs and GCD domains are integrally closed.
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Proof of Uncountable Basis for $\mathbb{N} \to \mathbb{R}$ over $\mathbb{R}$
Does anyone have a simple proof of the uncountability of bases of the vector space of all functions $f : \mathbb{N} \to \mathbb{R}$. I have seen a proof which uses the determinant of the Vandermonde matrix to show the linear independence of functions of the form $f_c=c^n$ but I believe that there might be a simpler one that doesn't require the use of matrices.
I have attached a link of another proof that I found online but I find the use of the limit unsettling. https://minhyongkim.wordpress.com/2013/10/23/a-vector-space-of-uncountable-dimension/
• Zorn's lemma gives you the existence of a basis. Not the fact that it is uncountable. Feb 12, 2017 at 14:29
• I was thinking along the lines of finding an uncountable linearly independent set of functions which implies that the bases cannot be countable. Feb 12, 2017 at 14:43
• And Zorn's lemma would help you how? Feb 12, 2017 at 14:53
• @AsafKaragila: I think he simply mean that Zorn's Lemma guarantees that every linearly independent set we find is a subset of a basis. Feb 12, 2017 at 14:56
Here is a diaginalisation argument.
Let $\{f_i\}$ be a countable set we find $g\not \in span \{f_i\}$.
Construct $g$ as follows.
Look at the vector $$(f_0(0), f_0(1))$$ define $(g(0),g(1))$ not to be a linear multiple of this vector. Now look at the vectors
$$(f_0(2), f_0(3), f_0(4))$$ $$(f_1(2), f_1(3), f_1(4))$$ define $(g(2),g(3),g(4))$ not a linear combination of these vectors. Now look at the vectors $$(f_0(5), f_0(6), f_0(7),f_0(8))$$ $$(f_1(5), f_1(6), f_1(7),f_1(8))$$ $$(f_2(5), f_2(6), f_2(7),f_2(8))$$
define $(g(5),g(6),g(7),g(8))$ not in the span of these three vectors, etc. I think the construction in clear, and $g$ is not in the span. Thus there is no countable basis. | {
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• That's a nice argument! Feb 12, 2017 at 14:49
• Sry i can't see how g is not in the span. Feb 12, 2017 at 14:50
• @JhonDeo: If it's in the span, then by definition is has to be in the span of finitely many of the $f_i$s. And there's a sequence of elements of $g$ that have been chosen explicitly not to be in the span of the first $n$ vectors, for any $n$. Feb 12, 2017 at 14:52
• @JhonDeo $g$ not in the span of $f_0$ since the first two coordinates of $g$ is not a multiple of the first two of $f_0$. $g$ is not in the span of $\{f_0,f_1\}$ since the coordinates $g(2),g(3),g(4)$ is not in the span or the same coordinates for $\{f_0,f_1\}$ etc. Feb 12, 2017 at 14:53
• @Henning Makholm, what does mean "not to be linear multiple of this vector"?
– ZFR
Nov 15, 2019 at 0:51
There are uncountably many functions $\mathbb N\to\{0,1\}$. Well-order them all, and then remove each one that is a (finite) linear combination of vectors that come earlier in the well-order. The result is, by contruction, a linearly independent set in your vector space. How large is it?
I claim that each of the removed vectors is not just a linear combination of vectors that remain, but a rational linear combination of such vectors. In each case, when we express the new vector as a linear combination of finitely many already-accepted ones, we have to solve an equation involving a matrix that is infinitely tall but has finite width. But because all elements are either $0$ or $1$, there are actually only finitely many different rows in the matrix, so we can find the coefficients by doing ordinary finite-dimensional linear algebra over $\mathbb Q$.
Now, if the reduced set of $0,1$-vectors were countable, there would be only countably many different finite rational combinations of them -- but this contradicts the fact that there are uncountably many vectors to express.
So we have an uncountable linearly independent set in your vector space, and if we extend that to a basis, we get an uncountable basis. | {
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I don't have a proof with Zorn's Lemma, but probably you like this one aswell:
The space you consider contains $\ell^\infty$, the space of bounded sequences, so if we can show that this space has an uncountable base the same is true for the space of all sequences.
The space $\ell^\infty$ can be made a Banach-space if we consider the supremum-norm on it. But infinite dimensional Banach-spaces can't have a countable Hamel-Base which is due to Baire's theorem.
If $(b_n)_{n\in\mathbb N}$ was a Hamel-base of $\ell^\infty$ then $\ell^\infty = \bigcup_{n\in\mathbb N}\operatorname{span}(b_1,...,b_n)$. Baire's theorem tells us that one of the sets in the union has nonempty interior, thus is all of $\ell^\infty$ because it is also a subspace. This is a contradiction because it would make $\ell^\infty$ finite-dimensional.
• Thanks I have seen this proof. However, I try to avoid such proofs as i have not learnt stuff like Banach Spaces and Hamel-base Feb 12, 2017 at 14:39
One easy way is the following, find a norm which makes the space complete and then uses baire theorem. Zorn's lemma implies that exist a basis, baire theorem implies it must be uncountable
• How do you find such a norm? Feb 13, 2017 at 6:17 | {
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# Mathematics
The value of a certain fraction becomes 1/5 if one is added to its numerator. If one is taken from its denominator, its value becomes 1/7. Find the fraction by setting up a pair of simultaneous equations and solving them
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2. 👎 0
3. 👁 171
1. (n+1)/d = 1/5
n/(d-1) = 1/7
5(n+1) = d
7n = d-1
5n - d = -5
7n - d = -1
2n = 4
n=2
d=15
Check:
n/d = 2/15
3/15 = 1/5
2/14 = 1/7
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2. 👎 0
posted by Steve
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# Systems of linear differential equations - eigenvectors
Solve the following system of equations
$\begin{cases} x_1^{'}(t)=x_1(t)+3x_2(t) \\ x_2^{'}(t)=3x_1(t)-2x_2(t)-x_3(t) \\ x_3^{'}=-x_2(t)+x_3(t)\end{cases}$.
First, I create the column vectors $X$ and $X^{'}$. Then the matrix $$A= \begin{bmatrix} 1 & 3 & 0 \\ 3 & -2 & -1 \\ 0 & -1 & 1 \\ \end{bmatrix}$$
Now, I find the eigenvalues, $-4,3,1$ and their corresponding eigenvectors $(-3,5,1)^T (-3,-2,1)^T (1,0,3)^T$.
I'm just not sure how to take it from here and solve the system of differential equations. I want a diagonal matrix $D$ so that I can read the solutions easy, but I'm not sure how to do it.
EDIT
Building on @Francisco 's answer, I'd have that:
$$X=c_1 (-3,5,1)^T e^{-4t} + c_2 (-3,-2,1)^T e^{3t} + c_3 (1,0,3)^T e^{t}$$. But I believe this could be written in a simpler form.
• Have you tried writing the matrix in a basis of its eigenvectors? (i.e in the form $S M S^{-1}$) where M is a diagonal matrix – Pol van Hoften Feb 28 '14 at 17:07
• To write $X^{'}=S^{-1}DSX$? – jacob Feb 28 '14 at 17:17
• You have $X'(t)=A X(t)$. Then $X=\vec v e^{\lambda t}$ (where $\vec v$ is an eigenvector and $\lambda$ is an eigenvalue) is a solution to the system. The general solution is given by $X=c_1 \vec v_1 e^{\lambda_1 t} +c_2 \vec v_2 e^{\lambda_2 t} + c_3 \vec v_3 e^{\lambda_3 t}$ where $c_1$ $c_2$ and $c_3$ are arbitrary constants. – Francisco Feb 28 '14 at 17:25
• What would be a faster approach? – jacob Feb 28 '14 at 17:56
• @Francisco I updated my question. Also, is there a wiki page for the formula you mentioned? – jacob Feb 28 '14 at 19:28 | {
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The matrix $A$ that you have is symmetric. So it has an orthonormal basis of eigenvectors. The eigenvectors you have found are mutually orthogonal (which they must be because they correspond to different eigenvalues.) So, if you normalize your eigenvectors and make those normalized vectors the columms of a matrix $U$, then $U^{T}U=I$ is automatic, and $U^{T}AU=D$ is diagonal. Explicitly, $$U = \left[ \begin{matrix} -\frac{3}{\sqrt{35}} & -\frac{3}{\sqrt{14}} & \frac{1}{\sqrt{10}} \\ \frac{5}{\sqrt{35}} & -\frac{2}{\sqrt{14}} & 0 \\ \frac{1}{\sqrt{35}} & \frac{1}{\sqrt{14}} & \frac{3}{\sqrt{10}} \end{matrix}\right]$$ The inverse of $U$ is the transpose $U^{T}$ of $U$. And, $$U^{T}AU = \left[\begin{matrix}-4 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 1\end{matrix}\right]=D.$$ Equivalently, $$A = U\left[\begin{matrix}-4 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 1\end{matrix}\right]U^{T}=U DU^{T}.$$ The general solution is expressed in terms of $a=x_{1}(0)$, $b=x_{2}(0)$, $c=x_{3}(0)$ as $$\left[\begin{matrix}x_{1}\\x_{2}\\x_{3}\end{matrix}\right] = e^{tA}\left[\begin{matrix}a\\b\\c\end{matrix}\right] = Ue^{tD}U^{T}\left[\begin{matrix}a \\ b \\ c\end{matrix}\right] = U\left[\begin{matrix} e^{-4t} & 0 & 0 \\ 0 & e^{3t} & 0 \\ 0 & 0 & e^{t}\end{matrix}\right]U^{T} \left[\begin{matrix}a \\ b \\ c\end{matrix}\right]$$
• This would be the same as $$\begin{cases} x_1= -3c_1 e^{-4t} - 3c_2e^{3t} +c_3e^{t} \\ x_2= 5c_1 e^{-4t} -2c_2e^{3t} \\ x_3= c_1 e^{-4t} + c_2e^{3t} +3c_3e^{t}. \end{cases}$$ yes? – jacob Feb 28 '14 at 20:55
• The forms are equivalent, except that the constants in mine are $x_{1}(0)$, $x_{2}(0)$, $x_{3}(0)$, while your constants are not that. Yours are $x_{1}(0)=-3c_{1}-3c_{2}+c_{3}$, etc. Mine will be the same if you set $a=-3c_{1}-3c_{2}+c_{3}$, $b=\dots$, etc. – Disintegrating By Parts Feb 28 '14 at 21:10 | {
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# Is $\int_0^\infty\frac{\arctan x}{\sqrt{x^3+x}}dx$ converges/diverges
I need to determine if the following integral converges/diverges.
$$\int_{0}^{\infty}\frac{\arctan x}{\sqrt{x^3+x}}dx$$
## What i tried:
We can write the integral as:
$$\int_{0}^{\infty}\frac{\arctan x}{\sqrt{x^3+x}}dx = \int_{0}^{\infty}\frac{\arctan x}{\sqrt{x(x^2+1)}}dx$$
Because the $$x^2+1$$ i thought about defining $$x = \tan u$$, therefore i will be able to use the identity: $$\tan^2u + 1 = \frac{1}{\cos^2u}$$
Define: $$x = \tan u \Rightarrow u = \arctan x$$ $$x = 0 \Rightarrow u = 0$$ $$x \to \infty \Rightarrow u = \pi/2$$
Therefore we can write the integral as:
$$\int_{0}^{\infty}\frac{\arctan x}{\sqrt{x(x^2+1)}}dx = \int_{0}^{\pi/2}\frac{u\cdot du}{\sqrt{\tan u(\tan^2u+1)}}$$
$$= \int_{0}^{\pi/2}\frac{u\cdot du}{\sqrt{\tan u\frac{1}{\cos^2u}}} = \int_{0}^{\pi/2}\frac{u\cos u}{\sqrt{\tan u}}du$$
And here i am stuck.
Another way:
I thought maybe to devide the integral into two, not sure even how, maybe:
$$\int_{0}^{\infty}\frac{\arctan x}{\sqrt{x^3+x}}dx = \int_{0}^{\pi/2}\frac{\arctan x}{\sqrt{x^3+x}}dx + \int_{\pi/2}^{\infty}\frac{\arctan x}{\sqrt{x^3+x}}dx$$
But i dont see how it gives me something.
Can i have a hint?
Thank you.
• Can't you also use the comparison test for this? Apr 6 '20 at 3:42
The long term behavior of the function is $$\frac{\pi/2}{x^{3/2}}.$$ The convergence of $$\int_1^{\infty} x^{-3/2} \, dx$$ should now tell you something. | {
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• I see what you say, when $x \to \infty$ we can write $arctanx$ as $\pi/2$ and the dominant variable at the denominator is $x^{3/2}$ but im not sure how to explain this so i can write my term as $\frac{\pi/2}{x^{3/2}}$ If i get to this, surly my integral converges
– Alon
Apr 6 '20 at 3:43
• Oh maybe i can write: $\frac{arctanx}{\sqrt{x^3+x}} < \frac{\pi/2}{\sqrt{x^3}}$
– Alon
Apr 6 '20 at 3:45
• Then use the comparison test
– Alon
Apr 6 '20 at 3:45
• @Alon Yes, that is a correct estimate, but that only solves half your problem; it ensures that $\displaystyle\int_1^{\infty} \dfrac{\arctan x}{\sqrt{x^3 + x}} \, dx \leq \displaystyle\int_1^{\infty} \dfrac{\pi/2}{x^{3/2}}\, dx < \infty$. But you also have to worry about $\displaystyle\int_0^1 \dfrac{\arctan x}{\sqrt{x^3 + x}}\, dx$ being finite. The answer by herb steinberg addresses this case. Apr 6 '20 at 5:27
• Thank you, i dont quite understand how i can say $\arctan x$ less or more equal $x$... We are in real analysis, can i say such things? its not... strict enough, isnt it?
– Alon
Apr 6 '20 at 5:30
As long as you are using the principal value for arctan$$x(\le \frac{\pi}{2}$$), then the integrand behaves like $$x^{-\frac{3}{2}}$$ as $$x\to \infty$$.
For $$x\to 0$$, arctan$$x \approx x$$, so the integrand behaves like $$\sqrt{x}$$.
Therefore you have convergence at both ends | {
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Therefore you have convergence at both ends
• Thank you, I dont understand how you formalaize behave like. for the deminator, you just say $\lim_{x \to 0}\sqrt{x^3+x} = \lim_{x \to 0}\sqrt{x^3}$ Is it a correct statment?
– Alon
Apr 6 '20 at 5:34
• For $x\to \infty$, $x^3\gg x$. For $x\to 0$, $x^3+x=x(x^2+1) \approx x$. Apr 6 '20 at 17:42
• But as far as i know this is an approximation you can write in something like physics, not, as much as i see, in real analysis, are you sure you can write such things in real analysis? If you do so great, i will take it
– Alon
Apr 6 '20 at 17:47
• The approximations are perfectly valid in real analysis. If you are taking a course where they first come up, you may have to formalize them. For example $x\lt x(x^2+1)\lt x(1+\epsilon)$ for $x\lt \sqrt{\epsilon}$ for any $\epsilon \gt 0$. Apr 6 '20 at 18:19
Note that for $$x>0$$, $$\arctan x < x$$ and $$\sqrt {x^3 + x} > \sqrt x$$, whence $$\frac{{\arctan x}}{\sqrt {x^3 + x}} < \sqrt x$$ which shows the convergence near $$0$$. Also for $$x>0$$, $$\arctan x < \frac{\pi }{2}$$ and $$\sqrt {x^3 + x} > \sqrt {x^3 } = x^{3/2}$$, whence $$\frac{{\arctan x}}{\sqrt {x^3 + x}} < \frac{\pi }{2}\frac{1}{{x^{3/2} }}$$ which shows the convergence at $$+\infty$$.
A quick way to show convergence is the limit comparison test.
For $$x \to 0^+$$ we have
$$\lim_{x\to 0^+}\frac{\arctan x}{x}=1 \text{ and } \lim_{x\to 0^+}\frac{\sqrt x}{\sqrt{x^3+x}} = 1 \Rightarrow \lim_{x\to 0^+}\frac{\frac{\arctan x}{\sqrt{x^3+x}}}{\sqrt x}= 1$$
Since $$\int_0^1 \sqrt x\; dx$$ is convergent, $$\int_0^1 \frac{\arctan x}{\sqrt{x^3+x}}dx$$ is convergent, as well.
For $$x \to +\infty$$ we have
$$\lim_{x\to +\infty}\left(\arctan x\cdot \frac{x^{\frac 32}}{\sqrt{x^3+x}}\right)=\frac{\pi}2\cdot 1\Rightarrow \lim_{x\to }\frac{\frac{\arctan x}{\sqrt{x^3+x}}}{\frac 1{x^{\frac 32}}}= \frac{\pi}2$$ | {
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Since $$\int_1^{+\infty} \frac{dx}{x^{\frac 32}}$$ is convergent, $$\int_1^{+\infty} \frac{\arctan x}{\sqrt{x^3+x}}dx$$ is convergent, as well. | {
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# Find the smallest value of n for which the nth term of the series is less than 0.001
Question:
The common ratio and the first term of a geometric series are 0.55 and 18 respectively.
Find the smallest value of n for which the nth term of the series is less than 0.001
$${\text{My solution: }}$$
$$Tn<0.001$$ $$ar^{n-1}<0.001$$
$$(18)(0.55)^{n-1}<0.001$$
$$(0.55)^{n-1}<\frac{1}{18000}$$ $$(n-l){\log 0.55}<{\log \frac{1}{18000}}$$ $$n-1<16.389$$ $$n<16.389+1$$ $$n<17.389$$ $$n=17$$
$${\text{However, the answer given for this question is}}$$ $$n=18$$ $${\text{Would anyone tell me either it is my answer or the answer given that is wrong ,please?}}$$ $${\text{Thank you.}}$$
## 1 Answer
$\log(0.55) < 0$ so from $$(n-1) \, \log(0.55) \lt \log \frac{1}{18000}$$ dividing (or multiplying) both sides by a negative number means your next line should be $$n-1 \gt 16.389\ldots$$ changing the direction of the inequality.
Checking the answers, $18 \times 0.55^{17-1} = 0.001262\ldots$ while $18 \times 0.55^{18-1} = 0.000694\ldots$ | {
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# Suppose that the average value on all intervals $[a,b]$ is equal to $f((a+b)/2)$. Prove that $f''(x) = 0$ for all $x \in \mathbb{R}$
I understand that $$f(x)$$ must be linear with a first derivative equal to a constant. I'm just not sure how I can use the mean value property of integrals to show something about $$f''(x)$$. The hint on this question is to use the fundamental theorem of calculus or Jensen's inequality.
• Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. – dantopa May 23 at 21:16
• You say that you understand that the first derivative must be a constant. Does this mean you have proven that part? I am not sure what you need help with. – InterstellarProbe May 23 at 21:17
Assuming $$f$$ is integrable and twice differentiable (otherwise your statement about average value doesn't make sense, nor your final statement*), $$\int_a^bf(x)\,\mathrm dx=(b-a)f\left(\cfrac{a+b}{2}\right)$$
Differentiate both sides w.r.t $$\,b$$, using the Leibniz integral rule (derived from fundamental theorem of calculus) for the LHS:
$$f(b)=\frac{b}{2}f'\left(\cfrac{a+b}{2}\right)+f\left(\cfrac{a+b}{2}\right)-\frac{a}{2}f'\left(\cfrac{a+b}{2}\right)$$
Now set $$b=0$$ and $$a=2x$$:
$$f(0)=f(x)-xf'(x)$$
Differentiate both sides w.r.t $$x$$:
$$0=f'(x)-f'(x)-xf''(x)$$
so $$f''(x)=0$$ for all $$x\neq 0$$.
Thus we have proven the function is linear everywhere except $$0$$. Since $$f'(0)$$ and $$f'(x)$$, $$f'(-x)$$ are constants for $$x>0$$ exists, we know $$f'(0)$$ has to be equal to each of these and thus $$f''(0)=0$$.
*I'm not sure if you're able to prove that $$f$$ has to be twice differentiable. | {
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*I'm not sure if you're able to prove that $$f$$ has to be twice differentiable.
• I see why $f'(x)$ and $f'(-x)$ are constant, because you've shown $f''=0$ for $x\neq 0$. Why does $f'(0)$ have to be equal to each of these constants? Is it because if it is not, then $f''$ is discontinuous at $x=0$? – Frudrururu May 24 at 4:11
• You can apply mean value theorem to prove that. In general, Darboux theorem tells that derivatives (if exist on an interval) enjoy intermediate value property regardless of it's continuity, and so, no jump discontinuity can occur to them. – Sangchul Lee May 24 at 6:10
I will assume that $$f$$ is locally integrable for an obvious reason. Our aim is to prove that $$f$$ is linear, which is then enough to conclude that $$f$$ is twice-differentiable with $$f'' \equiv 0$$.
Let $$x < y$$ and $$0 < \lambda < 1$$ be arbitrary. Set
$$c = \lambda x + (1-\lambda) y, \qquad a = 2x - c, \qquad b = 2y - c.$$
Then $$a < x < c < y < b$$ and
$$\frac{a+b}{2} = (1-\lambda)x + \lambda y, \qquad \frac{a+c}{2} = x, \qquad \frac{c+b}{2} = y.$$
So it follows that
\begin{align*} f((1-\lambda)x+\lambda y) &= \frac{\int_{a}^{b} f(t) \, \mathrm{d}t}{b-a} \\ &= \frac{\int_{a}^{c} f(t) \, \mathrm{d}t + \int_{c}^{b} f(t) \, \mathrm{d}t}{b-a} \\ &= \frac{(c-a)f(x) + (b-c)f(y)}{b-a} \\ &= (1-\lambda) f(x) + \lambda f(y). \end{align*}
This proves that $$f$$ is linear. | {
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# Proof by induction of Bernoulli's inequality $(1+x)^n \ge 1+nx$
I am working on getting the hang of proofs by induction, and I was hoping the community could give me feedback on how to format a proof of this nature:
Let $x > -1$ and $n$ be a positive integer. Prove Bernoulli's inequality: $$(1+x)^n \ge 1+nx$$
Proof:
Base Case: For $n=1$, $1+x = 1+x$ so the inequality holds.
Induction Assumption: Assume that for some integer $k\ge1$, $(1+x)^k \ge 1+kx$.
Inductive Step: We must show that $(1+x)^{k+1} \ge 1+(k+1)x$
Proof of Inductive Step: \begin{align*} (1+x)^k &\ge 1+kx \\ (1+x)(1+x)^k &\ge (1+x)(1+kx)\\ (1+x)^{k+1} &\ge 1 + (k+1)x + kx^2 \\ 1 + (k+1)x + kx^2 &> 1+(k+1)x \quad (kx^2 >0) \\ \Rightarrow (1+x)^{k+1} &\ge 1 + (k+1)x \qquad \qquad \qquad \square \end{align*}
-
Thanks for the suggestions, I'll keep your tips in mind. – KingOliver Aug 12 '12 at 15:09
Where did you uses $x>-1$? Hint: you did use it. – Thomas Andrews Aug 12 '12 at 15:16
When I claimed that $kx^2 >0$ – KingOliver Aug 12 '12 at 15:18
Actually, you need it when you multiply both sides by $1+x$. Also, since $x$ can be 0, $kx^2\ge0$ – Mike Aug 12 '12 at 16:06
Ah thank you @Mike – KingOliver Aug 12 '12 at 17:03
\begin{align*} (1+x)^{k+1}&=(1+x)(1+x)^k\\ &\ge(1+x)(1+kx)\\ &=1+(k+1)x+kx^2\\ &\ge1+(k+1)x\;, \end{align*}
since $kx^2\ge 0$. This completes the induction step.
This looks fine to me. Just a small note on formatting of the inequalities: I would combine the third and fourth inequalities as $$(1+x)^{k+1} \geq 1+(k+1)x+kx^2>1+(k+1)x,$$ so there is no need of the fifth line. Or even $$(1+x)^{k+1} = (1+x)(1+x)^{k} \geq (1+x)(1+kx)=1+(k+1)x+kx^2>1+(k+1)x.$$ | {
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# Homework Help: Solving for a variable in an equation that involves vectors
1. Apr 9, 2014
### Chetlin
1. The problem statement, all variables and given/known data
I have the equation:
$$\mathbf{F}_{2,1} = \frac{Q_1 Q_2}{4 \pi \varepsilon_0 {r_{2,1}}^2}\hat{r}_{2,1}$$ (standard electric force equation for 2 charges)
I know the value of everything except Q2 and have to find it. The vectors each have 3 components.
Normally in an algebraic equation, I would just solve for a variable by isolating it on one side of the = sign. But this equation involves vectors and I don't think there is a way to divide vectors. I could also subtract F2,1 from both sides which at least gets everything onto one side but I am still left with the vectors. I will effectively have three equations (one for each component of the vectors), but only one variable I have to solve for, right? Does the nature of this problem (it is physical) make it so that only certain vectors are even possible, so if I were to tweak F2,1 and not change the value of anything else, the problem would become impossible to solve because it would represent a physical impossibility?
2. Relevant equations
Nothing really
3. The attempt at a solution
I used only the x component of each vector and solved it using that equation and got the correct answer. But like I said before, if I tweaked only the y component of F2,1, the equation that used only the x components would be the same (so I'd get the same result) but the entire vector equation would not.
2. Apr 9, 2014
### ehild
r212 is not vector, but scalar. It is possible to divide with it. F2,1 is parallel with the unit vector $\hat{r}_{2,1}$ , a scalar multiple of it.
What were the data? You should determine r212 from the coordinates.
ehild
Last edited: Apr 9, 2014
3. Apr 9, 2014
### Chetlin | {
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ehild
Last edited: Apr 9, 2014
3. Apr 9, 2014
### Chetlin
That's true, but $\hat{r}_{2,1}$ is a vector, and I would want that to be on the other side as well. I just realized what I tried to do is impossible, because it would result in a scalar on one side of the = sign, and whatever you would (imaginatively) get if you divided two vectors, on the other side.
I'm starting to believe that any problem like this would be set up very specifically, and that only certain values will work at all, allowing me to just use one component of the vectors.
Edit: oops, I see you were still writing when I posted this.. sorry for jumping the gun so quickly, haha :P
Edit 2:
Oh dang, you're right. So you can't just tweak one component of F2,1.. doing that would prevent them from being parallel and would cause all kinds of issues. This is just a simple equation of a vector being equal to a magnitude times a direction unit vector. Sorry, it's been a little while since I've really worked with vectors in such a way, so I've been spending a lot of time on issues like this. Thanks again, very much!
Last edited: Apr 9, 2014
4. Apr 9, 2014
### ehild
You can not put $\hat{r}_{2,1}$ on the other side. But you know that $\vec{F}_{2,1}$ is a scalar multiple of $\hat{r}_{2,1}$, and that holds for all corresponding coordinates.
It would be better to see the whole text of the problem.
ehild
5. Apr 9, 2014
### Chetlin
Well, I have my question answered, but to be complete, here is the text of the problem. It's from an old Schaum's Outlines book from the early 1980's.
Point charge Q1 = 300 µC, located at (1, −1, −3) m, experiences a force F2,1 = (8, −8, 4) N due to point charge Q2 at (3, −3, −2) m. Determine Q2.
6. Apr 9, 2014
### Staff: Mentor
What you do is dot both sides of the equation by $\hat{r}_{2,1}$. This will give you the scalar equation you desire. Do you know how to determine $\hat{r}_{2,1}$?
Chet
7. Apr 9, 2014
### Chetlin | {
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Chet
7. Apr 9, 2014
### Chetlin
Yep, that unit vector is just the vector r2,1 divided by its magnitude ($\hat{r}_{2,1} = \frac{1}{r_{2,1}}\mathbf{r}_{2,1}$, where $r_{2,1} = ||\mathbf{r}_{2,1}|| = \sqrt{{r_{{2,1}_x}}^2 + {r_{{2,1}_y}}^2 + {r_{{2,1}_z}}^2}$). Sorry for the really weird notation inside the square root sign.
That's a really neat trick, thanks for showing it to me. I never thought to use scalar products to get rid of the vectors, and of course the scalar product of any unit vector with itself is 1.
8. Apr 10, 2014
### ehild
The point charge Q1 experiences force from Q2:
$$\vec F = k \frac {Q_1 Q_2}{(\vec r_1-\vec r_2)^2} \hat r_{12}$$
where $\hat r_{12}=\frac{\vec r_1-\vec r_2} {|\vec r_1-\vec r_2|}$
In your problem, r12=(1, −1, −3)-(3, −3, −2)=(-2, 2, -1). The magnitude is 3, so the components of $\hat r_{12}$ are (-2/3, 2/3, -1/3)
The force is F= (8, −8, 4) N. You see that the force is parallel to $\hat r_{12}$.
You can write out the Coulomb Law in x,y,z components:
$$F_x=8=\frac {kQ_1Q_2}{r_{12}^2}(-2/3)$$
$$F_y=-8=\frac {kQ_1Q_2}{r_{12}^2}(2/3)$$
$$F_z=4=\frac {kQ_1Q_2}{r_{12}^2}(-1/3)$$
You see that you get the same value for $kQ1Q2/ {r_{12}^2}$ from each equation, it is -12. kQ1Q2/9=-12. That is a scalar equation already.
ehild | {
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#1
How many positive integers less than $$1000$$ have the property that the sum of the digits of each such number is divisible by $$7$$ and the number itself is divisible by $$3$$?
Note by Vilakshan Gupta
1 year, 2 months ago
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Let's think of a number $$abc (0 \leq a, b, c \leq 9)$$. $$a+b+c \equiv 0 (\mod 3 \text{and} \mod 7)$$. Thus, $$a+b+c=21$$.
$(3, 9, 9) \rightarrow \frac{3!}{2}, (4, 8, 9) \rightarrow 3!, (5, 7, 9) \rightarrow 3!, (5, 8, 8) \rightarrow \frac{3!}{2}, (6, 6, 9) \rightarrow \frac{3!}{2}, (6, 7, 8) \rightarrow 3!, (7, 7, 7)$ $$3+6+6+3+3+6+1=28$$
Please tell me if there is any error.
- 11 months, 3 weeks ago
Good one brother
- 11 months, 3 weeks ago
Nice method
- 11 months, 3 weeks ago
33
- 2 months, 3 weeks ago
28
- 10 months, 2 weeks ago
27
- 1 year ago
its sum is divisibli by 21 using this you can solve
- 1 year ago
this question came in this year PRMO answer is 28
- 1 year ago
@Md Zuhair 8
- 1 year, 1 month ago
O i see....
- 1 year, 1 month ago
@Pokhraj Harshal Rajasthan region!
- 1 year, 1 month ago | {
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O i see....
- 1 year, 1 month ago
@Pokhraj Harshal Rajasthan region!
- 1 year, 1 month ago
And what about the other participants and their marks from your school. I mean the averages and the highest marks
- 1 year, 1 month ago
Which region are u from??
- 1 year, 1 month ago
WB rgion
- 1 year, 1 month ago
@Pokhraj Harshal Ok! Then I'm also getting the same...
- 1 year, 1 month ago
How much? Without the question?
- 1 year, 1 month ago
The questions will be cancelled
- 1 year, 1 month ago
Are the marks gonna be added to everyone's total or the questions will be cancelled (lowering the cutoff)?
- 1 year, 1 month ago
They will be added to totsl
- 1 year ago
Hey , what does discounted actually refer to?
- 1 year, 1 month ago
Hey I'm getting 8/27 from jharkhand as per the new answer key of hbcse.will I qualify???
- 1 year, 1 month ago
Lets see....
- 1 year, 1 month ago
Hello, There are 28 postive integers left less than 1000 have the property that the sum of the digits of each such number is divisible by 7 and the number itself is divisible by 3 You can check out for more queries related to the JEE EXAMS from the following compilation
- 1 year, 1 month ago
I believe the answer is 28 integers. The sum of these integers' digits must be divisible by 21, since a number divisible by 3 also has its sum of digits divisible by 3; in addition to the sum of digits divisible by 7. None of the digits can be less 3 since the sum of digits would be less than 21. Possible combinations = 7+6+5+4+3+2+1 = (7+1)+(6+2)+(5+3)+4=3*8+4=28.
- 1 year, 1 month ago
Ah - that's where the 28 comes from - much more mathematical than my just listing and counting them
- 1 year, 1 month ago
Exactly
- 1 year, 1 month ago
I agree with all that, and I got the same answer, but if I give you 4 digits at random (say 3, 4, 5 and 6) and ask how many numbers you can make out of them, the answer is 432*1 = 24, not 4+3+2+1 = 10. What am I missing?
- 1 year, 1 month ago | {
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- 1 year, 1 month ago
- 1 year, 2 months ago
Yup.I think so.
- 1 year, 2 months ago
@Md Zuhair @Vilakshan Gupta I haven't attempted one of the bonus question. Will I still get marks for it?
- 1 year, 2 months ago
I think the question can be cancelled as how can a person attempt to decinal answers and i had attempted ine. So i dunno.
- 1 year, 2 months ago
yeah hundreds digit cant be 1,2
- 1 year, 2 months ago
@Shreyan Chakraborty The Hundreds digit can't be 1 or 2..
- 1 year, 2 months ago
ANSWER IS 28....HAS A BIJECTION WITH a+b+c=21 WHERE 0<a,b,c<=9........
- 1 year, 2 months ago
Are na na.... I am not telling that. How much are you getting?
- 1 year, 2 months ago
@Shreyan Chakraborty .. How much?
- 1 year, 2 months ago
JANI NA BAJE HOYECHE
- 1 year, 2 months ago
LetTheFateDecide !!Bye
- 1 year, 2 months ago
which class are u in toshit?
- 1 year, 2 months ago
It's 11 in Rajasthan ! 😅😒
- 1 year, 2 months ago
According to Resonance , cutoff in Chandigarh is just 4(questions)
- 1 year, 2 months ago
i don't think it will be so low
- 1 year, 2 months ago
If that isnt, then wb will be higher and i will surely not qualify
- 1 year, 2 months ago
Ya. Thats ridiculous. WB region has always got a higher cutoff...
- 1 year, 2 months ago
Coz as per cutoff(s) uploaded by Resonance , cutoff in Chandigarh is lower than others ( Rajasthan , Maharashtra , UP , etc) ... That's why!
- 1 year, 2 months ago
So , you are already selected..Great 👍
- 1 year, 2 months ago
- 1 year, 2 months ago
oh
- 1 year, 2 months ago
Rajasthan..U?
- 1 year, 2 months ago
oh...btw,where do u live (i mean which region)
- 1 year, 2 months ago
Geometry was quite tough and lengthy! Excluding bonus , I'm getting 8
- 1 year, 2 months ago
Sir😅 I am getting 10 along with bonus!
- 1 year, 2 months ago
Oh. U mean 8/28 u r getting?
- 1 year, 2 months ago | {
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- 1 year, 2 months ago
Oh. U mean 8/28 u r getting?
- 1 year, 2 months ago
well, if it is bonus that means 2 questions marks are given extra.If it was been written question deleted then scores would be evaluated out of 28
- 1 year, 2 months ago
unfortunately, i will get only 10 questions correct. I did very silly mistakes
- 1 year, 2 months ago
@Md Zuhair Is the paper for 9,10,11 and 12 same?
- 1 year, 2 months ago
Yes Sir!
- 1 year, 2 months ago
How mañy have you got right in PRMO - 17?
- 1 year, 2 months ago
- 1 year, 2 months ago
we just need to find the numbers which add upto 21
- 1 year, 2 months ago
28
- 1 year, 2 months ago
25 is the answer as per me
- 1 year, 2 months ago
No - I get 28 too - I constructed a 0 - 9 by 0 -9 addition table in excel, and then started adding a 3rd digit to any number whose 2 digits had added to 12 or more - though now I think about it, I could just as easiky have srated my list with 399 and continued from there. And it has to be 28 cos it's one starting with3, 2 starting with 4, 3 starting with 5 etc, and 1+2+3+4+5+6+7 = 28
- 1 year, 2 months ago
What ans did you get?
- 1 year, 2 months ago
Hey Aaron. How much are u getting with bonus? With bonus i am getting 12.
- 1 year, 2 months ago
zuhair tui ki amk jiggesh korchish??
- 1 year, 2 months ago
Accha.. nijer whatsapp number ta de... whatsapp e kotha bolchi
- 1 year, 2 months ago | {
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# How many acres is 200 feet by 200 feet?
Jerilyn Kalandek asked, updated on October 26th, 2022; Topic: 200 feet
👁 511 👍 64 ★★★★☆4.9
200 feet multiplying by 200 feet equal to 40,000 sq ft. Therefore, 40,000 sq ft divide by 43,560 sq ft equal to 0.92 acres approximately.
Either way, what is the perimeter of a 1/4 acre?
There are 43560 square feet in an acre so one quarter of an acre is square feet. If each side of the square is F feet long then the area is F^2 square feet. Thus F^2 = 10890 and F = \sqrt{10890} = 104.35 feet.
Despite that, is an acre 200x200? An area 200 ft long by 200 ft wide encompasses 40,000 sq feet. This is 40000/43560 = 0.91827acres. 200 feet x 200 feet = 0.918 acres. Or in other words, approximately 92% of an acre.
Suitably, how many acres is 100 feet by 100 feet?
We know 43,560 square feet to 1 acre. 100 ft multiplying by 100 equal to 10,000 sq ft. Therefore, 10,000 sq ft divide by 43,560 sq ft equal to 0.23 acres approximately.
How much does a quarter acre cost?
2. Average Cost of Clearing Land by Lot Size
AcreageCost Range
¼ acre$125 -$1,400
½ acre$250 -$2,800
1 acre$500 -$5,600
2 acres$1,000 -$11,200
### What is the size of an acre lot?
1 acre is approximately 208.71 feet × 208.71 feet (a square) 4,840 square yards. 43,560 square feet.
### What is linear feet?
Technically, a linear foot is a measurement that is 12 inches long (so, one foot) and that is measured in a straight line, which is why it's called linear.
### What is a linear acre?
The most standard shape for an acre is one furlong by one chain, or 660 feet by 66 feet. To find the linear measurements of other rectangular acres, just divide 43,560 by the number of feet you want on one side. A square-shaped acre would then be about 208.7 by 208.7 feet (because 208.7 x 208.7 = ~43,560).
### How many acres is 150 feet by 200? | {
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### How many acres is 150 feet by 200?
We know 43,560 square feet to 1 acre. 200 feet multiplying by 200 feet equal to 40,000 sq ft. Therefore, 40,000 sq ft divide by 43,560 sq ft equal to 0.92 acres approximately.
### How many square feet is 200?
We know about 43,500 square feet. 20,000 sq ft is equal to 200 feet. 43,500 sq ft is equal to 0.92 sq ft.
### How big is half an acre compared to a football field?
If you calculate the entire area of a football field, including the end zones, it works out to 57,600 square feet (360 x 160). One acre equals 43,560 square feet, so a football field is about 1.32 acres in size.
### How big is a 50 by 100 plot of land?
1)For a rectangular plot, 50 by 100 refers to 50 feet by 100 feet which is equivalent to 15 meters by 30 meters and is also equal to 450 square meters. This is what people refer to as 1/8 of an acre though slightly less due to the provision for access road.
### What does 1acre look like?
As all farmers and real estate agents know, an acre is defined as an area one furlong long by 4 rods wide. ... Basically if you can picture a football field, that's pretty close to an acre in size. Officially, it is 43,560 square feet, and a football field is 48,000 square feet. Our standard acre isn't the same worldwide.
### How long does it take to walk around 4 acres?
A square acre is 208.7 feet on a side, so the perimeter of an acre is about 835 feet, or about 16 percent of a mile. If you walk a brisk pace of 3 miles an hour, you can cover a mile in 20 minutes. So you should be able to walk 835 feet in about three minutes.
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# Modular Arithmetic - Are we allowed to distribute the Modularity?
Assume I have a problem such as "Prove that $\displaystyle103^{53} + 53^{103}$ is divisible by $39$." This would mean I wanted to prove that $\displaystyle103^{53} + 53^{103}\equiv0\pmod{39}$.
My starting statement would be then "$\displaystyle103^{53} + 53^{103}\pmod{39}$" and I would then equate this to "$\displaystyle103^{53}\pmod{39} + 53^{103}\pmod{39}$" and then continue.
Am I allowed to distribute the mod like that?
Thanks.
-
Yes, you are... – DonAntonio Mar 7 '14 at 16:06
@Matthew, please verify the edited version – lab bhattacharjee Mar 7 '14 at 16:14
That should be a $+$ rather than a comma, right? In the formula right before "and then continue". – Jack M Mar 7 '14 at 17:38
Mathematicians and programmers have two different ways of thinking about mod. To the latter, it is a binary operation in which ${\rm Mod}[a,b]$ is the remainder of $a$ when divided by $b$ (so it will return an integer in the range $0\le r<b$). To the former, it is a binary relation (a symbol that relates things in some way, like $<,=,>,\approx,\sim$ etc.) for each modulus $b$. We say that $n\color{Red}{\equiv}m$ mod $b$ if the difference $n-m$ is divisible by $b$, or equivalently if ${\rm Mod}[n,b]={\rm Mod}[m,b]$. Sometimes the equivalence symbol $\equiv$ is simply replaced by an equality symbol $=$, in which case we are understood to be equating equivalence classes. The relation $n\equiv m$ mod $b$ is in fact a congruence relation (it "respects" $+$ and $\times$), and the equivalence classes are called residue classes, or just residues.
So if $a\equiv b$ and $c\equiv d$ mod $m$ then $ac\equiv bd$ and $a+c\equiv b+d$ mod $m$. One can use this to end up proving that, in particular, $f(a)\equiv f({\rm Mod}[a,m])$ mod $m$ for integer-coefficient polynomials $f$. | {
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Thus for example ${\rm Mod}[a,m]+{\rm Mod}[b,m]$ and $a+b$ and ${\rm Mod}[a+b,m]$ are all congruent mod $m$, however it is not strictly true that the first and last are equal as integers. Take $a,b=2$ and $m=3$, in which case ${\rm Mod}[2,3]+{\rm Mod}[2,3]=2+2=4\ne1={\rm Mod}[2+2,3]$, albeit $4\equiv1$ mod $3$.
If $n$ and $m$ are coprime, then $a\equiv b$ modulo both $n$ and $m$ if and only if $a\equiv b$ mod $nm$. In particular this means $x\equiv 0$ mod $39$ if and only if $x\equiv0$ mod $3$ and mod $13$. Compute
$$103^{53}+53^{103}\equiv 1^{53}+(-1)^{103}\equiv 1+(-1)\equiv0\mod 3$$
because $103\equiv1$ and $53\equiv-1$ mod $3$. And then compute
$$103^{53}+53^{103}\equiv(-1)^{53}+1^{103}\equiv(-1)+1\equiv0\mod 13$$
because $103\equiv-1$ and $53\equiv1$ mod $13$. Since $103^{53}+53^{103}$ is $0$ mod $3$ and $13$, it is $0$ mod $39$.
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Good, and complete. For the purposes of students going farther in mathematics, it is disastrous to keep the mindset of relying on the operation rather than the equivalence relation. When talking about $R/I$ where $R$ is a ring and $I$ is an ideal, or $G/H$ where $\supset H$ are groups, the equivalence relation outlook is essential. – Lubin Mar 7 '14 at 18:49
If $103^{53} \equiv a \mod 39$ and $53^{103} \equiv b \mod 39$, than it is indeed true that $103^{53} + 53^{103} \equiv a+b \mod 39$. I wouldn't recommend your notation, which makes $\mod\mbox{ }$ look like an operator. | {
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There's nothing wrong with using $\bmod$ as an operator. – ShreevatsaR Mar 7 '14 at 16:54
Using mod as an operator loses the properties of the equivalence class that makes it so convenient. For example, you cannot assume $a \mod M + b \mod M = (a + b) \mod M$. – DanielV Mar 7 '14 at 17:48
@DanielV: Assuming the name "mod" refers to a proper modulus operator and not the much-less-useful "remainder" operator found in many programming languages, and one is not expecting one's numeric type to act as an algebraic ring with a modulus which is not a multiple of M, why can't one assume that (a mod M)+(b mod M) will equal ((a+b) mod M) in cases where the result of the addition is defined? – supercat Mar 7 '14 at 18:25
Ah, if I understand, you are talking about a $\mathbb{N} \rightarrow \text{Galois}[M]$ operator, yes that would be useful indeed. – DanielV Mar 8 '14 at 0:02
To say that $$x \equiv a \pmod{m}$$ means that there is some integer $k$ such that $$x - a = mk$$ Thus, expanding on user133281's answer (i.e. if $103^{53}\equiv a \pmod{39}$ and $53^{103}\equiv b \pmod{39}$), we have $$(103^{53}-a)+(53^{103}-b)=39k+39l$$ which is equivalent to $$(103^{53}+53^{103})-(a+b)=39(k+l)$$ so $$103^{53}+53^{103}\equiv a+b\pmod{39}$$
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Either you meant $\mod 39$ in the first line or $mk$ in the second. – Mike Miller Mar 8 '14 at 3:45
@Mike Right you are. Thank you. – iamnotmaynard Mar 8 '14 at 20:56
Glad to help!$\$ – Mike Miller Mar 8 '14 at 21:37
HINT:
Please have a look into this for the properties of congruence
$\displaystyle 103\equiv-1\pmod{13}\implies 103^{53}\equiv(-1)^{53}\equiv-1$
$\displaystyle 53\equiv1\pmod{13}\implies 53^{103}\equiv(1)^{103}\equiv1$
Similarly, for $\pmod3$
Now if $13$ and $3$ both divides $a,a$ will be divisible by lcm$(13,3)$ | {
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Similarly, for $\pmod3$
Now if $13$ and $3$ both divides $a,a$ will be divisible by lcm$(13,3)$
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While this is a good hint for solving the problem "show that $103^{53} + 53^{103}$ is divisible by 39", this doesn't address the question that was asked ("am I allowed to distribute the mod") at all. – Magdiragdag Mar 7 '14 at 16:05
@Magdiragdag, added a relevant link – lab bhattacharjee Mar 7 '14 at 16:10
The "equivalent modulo $n$" relation is a congruence: a congruence is an equivalence relation with the additional property that for all relevant arithmetic operations (in this case, $0, 1, +, \times$, and anything derived from those), if the inputs the the arithmetic operation are congruent, then the outputs are also congruent.
In the actual ring of integers modulo $n$, though, there is no "congruence" or "mod": e.g. $39 = 0$ is a literal equality. Your question to prove
$$103^{53} + 53^{103} \equiv 0 \pmod{39}$$
in the integers is the same thing as trying to prove
$$103^{53} + 53^{103} = 0$$
in the ring of integers modulo $39$. And since we're working with actual equality, it's clear that we can simplify the two summands separately, then add the simplified results.
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# Prove that the expression is divisible by 26460
#### anemone
##### MHB POTW Director
Staff member
Problem:
Prove that $27195^8-10887^8+10152^8$ is divisible by $26460$.
Attempt:
I grouped the last two terms and manipulated them algebraically and came to the point where I suspect I might have taken the wrong path...here is the last step where I stopped and don't know how to proceed.
$\dfrac{27195^8-10887^8+10152^8}{26460}=\dfrac{3^5\cdot5^7\cdot7^{14}\cdot37^8-7013(2^6\cdot3^6\cdot47^2+3^2\cdot19^2\cdot191^2)(10152^4+10887^4)}{8}$
I'd like to ask, do you think this problem can be solved using only elementary methods?
Last edited by a moderator:
#### Opalg
##### MHB Oldtimer
Staff member
Problem:
Prove that $27195^8-10887^8+10152^8$ is divisible by $26460$.
Attempt:
I grouped the last two terms and manipulated them algebraically and came to the point where I suspect I might have taken the wrong path...here is the last step where I stopped and don't know how to proceed.
$\dfrac{27195^8-10887^8+10152^8}{26460}=\dfrac{3^5\cdot5^7\cdot7^{14}\cdot37^8-7013(2^6\cdot3^6\cdot47^2+3^2\cdot19^2\cdot191^2)(10152^4+10887^4)}{8}$
I'd like to ask, do you think this problem can be solved using only elementary methods?
The best way to make the calculation more manageable is to factorise $26460 = 2^2\cdot 3^3\cdot 5\cdot 7^2$. If you can separately show that $27195^8-10887^8+10152^8$ is divisible by each of the numbers $2^2$, $3^3$, $5$ and $7^2$, then the result will follow.
Take the factor 5, for example. Since $27195$ is a multiple of $5$, so is its eighth power. The other two terms are not multiples of $5$, but here you need to use your idea of grouping those two terms together. In fact, $a^8-b^8$ is a multiple of $a-b$. So $10887^8+10152^8$ is a multiple of $10887-10152 = 735$. That is a multiple of $5$. Putting those results together, you see that $27195^8-10887^8+10152^8$ is a multiple of $5$. | {
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The exact same procedure shows that $27195^8-10887^8+10152^8$ is a multiple of $7^2$. You can use similar ideas to show that it is also a multiple of $3^3$ and of $2^2$. (To deal with $3^3$, notice that $a^8-b^8$ is also a multiple of $a+b$.)
#### anemone
##### MHB POTW Director
Staff member
The best way to make the calculation more manageable is to factorise $26460 = 2^2\cdot 3^3\cdot 5\cdot 7^2$. If you can separately show that $27195^8-10887^8+10152^8$ is divisible by each of the numbers $2^2$, $3^3$, $5$ and $7^2$, then the result will follow.
Take the factor 5, for example. Since $27195$ is a multiple of $5$, so is its eighth power. The other two terms are not multiples of $5$, but here you need to use your idea of grouping those two terms together. In fact, $a^8-b^8$ is a multiple of $a-b$. So $10887^8+10152^8$ is a multiple of $10887-10152 = 735$. That is a multiple of $5$. Putting those results together, you see that $27195^8-10887^8+10152^8$ is a multiple of $5$.
The exact same procedure shows that $27195^8-10887^8+10152^8$ is a multiple of $7^2$. You can use similar ideas to show that it is also a multiple of $3^3$ and of $2^2$. (To deal with $3^3$, notice that $a^8-b^8$ is also a multiple of $a+b$.)
Awesome! I finally understand it now!
#### Bacterius
##### Well-known member
MHB Math Helper
Opalg's answer is very good! A more "heavy machinery" method could also go as follows:
$$26460 = 2^2 \cdot 3^3 \cdot 5^1 \cdot 7^2$$
And we can do the following reductions with a couple modulo operations:
$$27195^8 - 10887^8 + 10152^8 \equiv 3^8 - 3^8 + 0^8 \equiv 0 \pmod{2^2}$$
$$27195^8 - 10887^8 + 10152^8 \equiv 6^8 - 6^8 + 0^8 \equiv 0 \pmod{3^3}$$
$$27195^8 - 10887^8 + 10152^8 \equiv 0^8 - 2^8 + 2^8 \equiv 0 \pmod{5^1}$$
$$27195^8 - 10887^8 + 10152^8 \equiv 0^8 - 9^8 + 9^8 \equiv 0 \pmod{7^2}$$
And invoking the CRT (since $2^2$, $3^3$, $5^1$, $7^2$ are pairwise coprime): | {
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$$\mathbb{Z}_{26460} = \mathbb{Z}_{2^2} \times \mathbb{Z}_{3^3} \times \mathbb{Z}_{5^1} \times \mathbb{Z}_{7^2}$$
$$\therefore$$
$$27195^8 - 10887^8 + 10152^8 \equiv 0 \pmod{26460}$$
This also shows that the exponent (8) is in fact irrelevant, and could be anything.
EDIT: fixed, see ILikeSerena's post below.
Last edited:
#### Klaas van Aarsen
##### MHB Seeker
Staff member
Opalg's answer is very good! A more "heavy machinery" method could also go as follows:
$$26460 = 2^2 \cdot 3^3 \cdot 5^1 \cdot 7^2$$
And we can do the following reductions with a couple modulo operations:
$$27195^8 - 10887^8 + 10152^8 \equiv 0^8 - 0^8 + 0^8 \equiv 0 \pmod{2}$$
$$27195^8 - 10887^8 + 10152^8 \equiv 0^8 - 0^8 + 0^8 \equiv 0 \pmod{3}$$
$$27195^8 - 10887^8 + 10152^8 \equiv 0^8 - 2^8 + 2^8 \equiv 0 \pmod{5}$$
$$27195^8 - 10887^8 + 10152^8 \equiv 0^8 - 2^8 + 2^8 \equiv 0 \pmod{7}$$
And invoking the CRT (since 2, 3, 5, 7 are pairwise coprime):
$$\mathbb{Z}_{26460} = \mathbb{Z}_{2^2} \times \mathbb{Z}_{3^3} \times \mathbb{Z}_{5^1} \times \mathbb{Z}_{7^2}$$
$$\therefore$$
$$27195^8 - 10887^8 + 10152^8 \equiv 0 \pmod{26460}$$
This also shows that the exponent (8) is in fact irrelevant, and could be anything.
I'm afraid that the number $2 \cdot 3 \cdot 5 \cdot 7$ is also zero mod 2, mod 3, mod 5, and mod 7.
But it is not divisible by 26460.
For CRT you need to show the mod relation for $2^2$, $3^3$, $5^1$, and $7^2$.
For instance for $3^3$, we need:
$\hspace{0.5 in}27195^8 - 10887^8 + 10152^8 \equiv 0^8 - 0^8 + 0^8 \equiv 0 \pmod{3^3}$
Still quite doable.
#### Bacterius
##### Well-known member
MHB Math Helper
I'm afraid that the number $2 \cdot 3 \cdot 5 \cdot 7$ is also zero mod 2, mod 3, mod 5, and mod 7.
But it is not divisible by 26460.
For CRT you need to show the mod relation for $2^2$, $3^3$, $5^1$, and $7^2$.
For instance for $3^3$, we need:
$\hspace{0.5 in}27195^8 - 10887^8 + 10152^8 \equiv 0^8 - 0^8 + 0^8 \equiv 0 \pmod{3^3}$ | {
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$\hspace{0.5 in}27195^8 - 10887^8 + 10152^8 \equiv 0^8 - 0^8 + 0^8 \equiv 0 \pmod{3^3}$
Still quite doable.
True, my mistake. Though it's not too difficult to check, just do the same operations but using 4, 27 and 49... (which still work out, it's clear the integers in the problem were carefully chosen to cancel each other out) | {
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# Alternative ways to find $\lim_{x\to 1}\; \frac{1-x}{1- \sqrt{x}}$
Searching on Google by a solution, I found this:
$1) \space$The numerator can be written by a diference of squares:$\space (1-\sqrt{x})(1+\sqrt{x})$
$2 \space)$Then, one can "eliminated" the commum factor beteween the numerator and the denominator.
$3) \space$The final expression looks like: $$\lim_{x\to 1}\; {1+ \sqrt{x}}=2$$
However this was not a intuitive algebraic solution. I haven't thought of this solution at the first attempts. Could you please give me other alternative algebraic solutions, if it exists. Thanks.
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Same solution, let $x=u^2$, but now instantly familiar. – André Nicolas Jan 7 '13 at 17:20
Why isn't it intuitive? If you have a $0/0$ form, the first thing one usually does is search for common factors. – David Mitra Jan 7 '13 at 17:21
The difference of two squares method was the first thing that popped into my head before I clicked the link to this question! – Clive Newstead Jan 7 '13 at 17:21
The other approach to this solution is to think - that was neat, how will I be able to spot neat solutions like that next time and the comment from @AndréNicolas was exactly what I was going to put. – Mark Bennet Jan 7 '13 at 17:22
@João What's $(1-\sqrt{x})(1+\sqrt{x})$? – WimC Jan 7 '13 at 17:35
You can compute the inverse: $$\lim_{x\to 1}\frac{1-\sqrt{x}}{1-x}=\lim_{x\to 1}\frac{\sqrt{x}-1}{x-1}$$ which is, by definition, $f'(1)$ where $f(x)=\sqrt{x}$.
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As I understood, you "turn over" the fraction,by puting $\frac{1-\sqrt{x}}{1-x}$ on the denominator. Then you multiplied the denominator by $\frac{-1}{-1}$ to get the final expression, that is under the main fraction (the numerator is $1$).Finally you computed the derivative of $\sqrt{x}$ at $x=1$ to get $\frac{1}{\frac{1}{2}}$. Is that? – João Jan 7 '13 at 18:04
Yeah, that's basically it. Essentially, if $g(x)/h(x)\to A$ then $h(x)/g(x)\to 1/A$ – Thomas Andrews Jan 8 '13 at 14:12 | {
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Same solution, let $x=u^2$, but now instantly familiar.
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Perhaps we can use L'Hospital rule here ?
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By substitution:$\quad$ let $x = u^2\,.\quad$Then as $x\to 1,\;\;u^2\to 1\implies u\to 1$, giving us $$\lim_{x\to 1}\; \frac{1-x}{1- \sqrt{x}} \;\;=\;\; \lim_{u\to 1}\; \frac{1-u^2}{1- u} = \lim_{x\to 1} \frac{u^2 - 1}{u-1} \;=\;\lim_{u \to 1} \frac{(u-1)(u+1)}{u-1} = \lim_{u\to 1}(u + 1) = 2$$
Using the substitution makes the "difference of squares" route looks so much more obvious!
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If $u^2 \to 1$, then $u \to 1$ or $u \to -1$. Is this second situation if one compute the limit we get $0$.M'I rigth? – João Jan 7 '13 at 18:14
since $x \to 1$, the only root of concern is u = 1 (since $u = \sqrt{x}$ must necessarily be $\ge 0$ in this situation. – amWhy Jan 7 '13 at 18:22 | {
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# Why $1\equiv a^{p-1} \mod p$?
Let $\mathbb{Z}_p^*$, where $p$ is prime and let $a\in\mathbb{Z}_p^*$.
Consider the following equation:$$(p-1)! \equiv (p-1)! a^{p-1} \mod p$$
I've read that since $\gcd((p-1)!, p) = 1$ we can infer that $$a^{p-1} \equiv 1$$
So I have two questions:
1. Why is it true that $\gcd ((p-1)!, p)= 1$?
2. Why can we infer that $a^{p-1} \equiv 1$?
• for the frist question: if $p$ is prime then observe than $(p-1)!$ doesnt contain $p$ as a factor, hence $\gcd((p-1)!,p)=1$ – Masacroso Mar 19 '17 at 15:44
• And for the second question, you can always write the $gcd(a,b)$ as a linear combination of $a$ and $b$ with integer coefficients (this is from the Euclidean algorithm). Use this to show that if some number is relatively prime to $p$, then it has a multiplicative inverse mod $p$. – Malkoun Mar 19 '17 at 15:46
• Fermat's little theorem en.wikipedia.org/wiki/Fermat%27s_little_theorem?wprov=sfla1 – Zuo Mar 19 '17 at 15:51
1. The only divisors of $p$ are $p$ and $1$. What are the divisors of $(p-1)!$?
2. Once you know that $p, (p-1)!$ are relatively prime, consider $(p-1)! * (a^{p-1} - 1) \equiv 0 \mod p$. The relative primeness tells you that this is only possible if $(a^{p-1} - 1) \equiv 0 \mod p$
This common result is known as Fermat's Little Theorem. I hope this simple proof helps:
Consider the sequence of integers $n,2n,3n,…,(p−1)n$.
Note that none of these integers are congruent modulo $p$ to the others.
If this were the case, we would have $an≡bn \pmod p$ for some $1≤a<b≤p−1$.
Then as $gcd(n,p)=1$, and we can cancel the $n$, we get $a≡b \pmod p$ and so $a=b$.
Also, since $p∤n$ and $p∤c$, for any $1≤c≤p−1$, then by Euclid's Lemma $p∤cn$ for any such $cn$, which means $cn≢0 \pmod p$.
Thus, each integer in the sequence can be reduced $modulo \ p$ to exactly one of $1,2,3,…,p−1$.
So ${1,2,3,…,p−1}$ is the set of Reduced Residue System $modulo \ p$. | {
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So ${1,2,3,…,p−1}$ is the set of Reduced Residue System $modulo \ p$.
So, upon taking the product of these congruences, we see that $n×2n×3n×⋯×(p−1)n≡1×2×3×⋯×(p−1) \mod p$.
This simplifies to $n^{p−1}×(p−1)!≡(p−1)! \pmod p$.
Since $p∤(p−1)!$, we can cancel $(p−1)!$ from both sides, leaving us with $n^{p−1}≡1 \pmod p$.
• The last statement is essentially my question. Why does the fact $p\not\vert (p-1)!$ implies that you can cancel $(p-1)!$ from both sides? – OliOliver Mar 19 '17 at 16:14
• It doesn't essentially imply that $(p-1)!$ is being cancelled because of that property. It merely states that they are relatively prime to each other and, hence, we can use the property of modular arithmetic to basically remove $(p-1)!$ as if it of no use here(as we are dealing with, say, modulo $p$ and not modulo $(p-1)!$) and appears on both sides leaving it to be factored out. – HKT Mar 19 '17 at 16:25
• Carefully think about it. Suppose if $p|(p-1)!$, we would have been left with $0 \pmod p$ on the right hand side. I hope that helps. – HKT Mar 19 '17 at 16:28
• You are just over-thinking it. I assure you once you get the hang of it and review the basic modular arithmetic definitions and get a stronger grasp on the notation, there will be clarity. – HKT Mar 19 '17 at 16:30
If you write down $(p-1)!$ as $(p-1)\cdot(p-2)\cdot\cdot\cdot1$ you can easily notice that $p$ and $(p-1)!$ have no common factors.
So $gcd((p-1)!,p) = 1$.
Now, knowing that:
$$ax \equiv b \mod(m) \implies x \equiv b \cdot a^{-1} \mod(\frac{m}{gcd(m,a)})$$ | {
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$$ax \equiv b \mod(m) \implies x \equiv b \cdot a^{-1} \mod(\frac{m}{gcd(m,a)})$$
• You might as well just write $ax\equiv b\bmod m\implies x=a^{-1}b\bmod m$ when $\gcd(a,m)=1$. No need to overcomplicate it. – arctic tern Mar 19 '17 at 15:55
For the first question, take, for example, $p = 7$. Then $(p-1)! = 2\cdot3\cdot4\cdot5\cdot6 = 2^4\cdot3^2\cdot5$. Notice that in the first equation all of the factors are strictly less than $p = 7$, which implies that all of the prime factors are also strictly less than $p$. Since $p$ is prime, this means that $p$ and $(p-1)!$ can have no prime factors in common.
Now this implies Fermat's little theorem because, having shown that $(p-1)!$ is relatively prime with $p$, we can apply the cancellation law.
Whenver $\gcd(a,m) = 1$, $$ax \equiv ay \mod m$$ $$\downarrow$$ $$x \equiv y \mod m$$ | {
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Rolling two dice alternatively. What is probability of winning?
Tom and Paul roll 2 dice alternatively starting with Tom. Consider they use two fair 6-faced dice. The player who rolls 6 first wins. They continue to roll until one of them wins. Find probability that Tom wins.
I have listed out the total possible outcomes below:
{(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
Here are all the possible outcomes of two dice rolls. And as one can see, only 5 of these comes out with a 6 first; so, I think the probability of Tom winning should be 5/36.
However, when I looked at the solution (This is a question from one midterm in one of previous semesters given at my school) says:
p is 5/36
$$P(Tom\, Wins)=\sum_{k=0}^{\infty }p(1-p)^{2k}=p\sum_{k=0}^{\infty }(1-p)^{2k}=$$ $$\frac{\frac{5}{36}}{1-\frac{31}{36}}=\frac{5}{5}$$
As one can see this answer is really rediculus! (Tom will win no matter what??)
I think that the key here is probably on the word "alternatively", but cannot figureout what has gone wrong here.
• You’ve computed that sum incorrectly. Observe that the exponent is $2k$, not $k$, i.e., you’ve got a geometric series in $(1-p)^2$, not $1-p$.
– amd
Apr 3 '20 at 19:50
• I didn't come out with that answer. It is answer given by someone. I looked at it and was very confused. Apr 4 '20 at 1:37 | {
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The probability that Tom wins on the first roll is $$\frac16$$. He wins on the third roll if and only if he first rolls something other than a $$6$$, then Paul rolls something other than a $$6$$, and then Tom rolls a $$6$$; the probability of this is $$\frac56\cdot\frac56\cdot\frac16$$. In general, Tom wins on the $$(2k+1)$$-st roll if and only if the first none of the first $$2k$$ rolls is a $$6$$, and Tom rolls a $$6$$ on roll $$2k+1$$; this occurs with probability $$\frac16\left(\frac56\right)^{2k}$$. Thus, the probability that Tom wins is
\begin{align*} \sum_{k\ge 0}\frac16\left(\frac56\right)^{2k}&=\frac16\sum_{k\ge 0}\left(\frac56\right)^{2k}\\ &=\frac16\sum_{k\ge 0}\left(\frac{25}{36}\right)^k\\ &=\frac16\cdot\frac1{1-\frac{25}{36}}\\ &=\frac6{11}\;. \end{align*}
This is the formula in the solution that you read, with $$p=\frac16$$.
It should not be surprising that this is slightly more than $$\frac12$$: the fact that Tom goes first gives him an advantage, but it’s a small one, since the game is otherwise very symmetric.
One can also compute the desired probability without resort to infinite series. Let $$p$$ be the probability that Tom wins, so that Paul wins with probability $$1-p$$. On the other hand, the probability that Paul when Tom first rolls something other than a $$6$$ must be $$p$$, because at that point the game is effectively starting over with Paul as the first player. Thus, Paul wins with probability $$\frac56p$$, the probability that Tom rolls something other than a $$6$$ initially and Paul then rolls a $$6$$ before Tom does. In short, $$1-p=\frac56p$$, and solving for $$p$$ again yields $$p=\frac6{11}$$. | {
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• Thank you. I think your answer is good. I guess the solution given for this problem is probably wrong. Apr 4 '20 at 1:36
• @AmosKu: You’re welcome. Apr 4 '20 at 1:42
• I have given the question quite a bit of thought last night. I think the key is "alternatively". And whenever Tom rolls a 6, he wins. I think your answer is for the case where only one 6 face dice is rolled. I have edited my wording above, hoping to make the meaning of original question clearer. Apr 4 '20 at 17:44
• @AmosKu: Alternatively makes no sense here; I suspect that the intended word is alternately. That simply means that Tom rolls his die, then Paul rolls his, and so on. This is exactly the same problem as if they were using a single die and trading it back and forth. Apr 4 '20 at 17:52
• Possible. But I just double checked the original question. The word is " Alternatively". There are other errors in the writing of the test as well. I guess you are right. Apr 4 '20 at 20:07 | {
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# Conclusion about the dimension of C°(R)?
Tags:
1. Jul 12, 2017
### Rodrigo Schmidt
[mentor note: thread moved from Linear Algebra to here hence no homework template]
We have a linear map A:E→E, where E=C°(ℝ), the vector space of all continuous functions.
Let's suppose that Aƒ= x0 ƒ(t)dt.
By the Calculus Fundamental Theorem, d/dx(Aƒ) = ƒ, so we have a left inverse, which implies that ker(A)={0}.
Supposing that dim(E) is finite, by the rank-nullity theorem we have that im(A)=E . As a result of that:
(ƒ(x)=|x|) ∈ E ⇒ ƒ ∈ im(A)
⇒dƒ/dx ∈ E
But we know that ƒ's derivative is not continuous. So, supposing that dim(E) is finite lead us to a contradition (dƒ/dx ∈ E ∧ dƒ/dx ∉ E) therefore dim(E) must be infinite.
Is this argument valid? If not, could you guys point where does it fail? Thank you!
Last edited by a moderator: Jul 12, 2017
2. Jul 12, 2017
### Krylov
Note that the right-hand side is $(Af)(x)$, not $Af$.
The derivative is not even defined at $x = 0$.
Yes, it is valid, but it is quite convoluted. If you insist on doing it this way, I would prefer to say that you have found a linear operator $A$ that is injective but not surjective (the latter because $A$ maps into $C^1(\mathbb{R})$). This already implies infinite dimensionality of $E$, since on a finite dimensional space injectivity and surjectivity are equivalent for linear operators.
Of course, if you merely care to show that $E$ is infinite dimensional, it is more straightforward to identify an infinite linearly independent set.
3. Jul 12, 2017
### Rodrigo Schmidt
Thanks for the knowledge shared!
I hadn't tought that, that's, indeed, much simpler.
So the basis of the set of all polynomials would be enough?
4. Jul 12, 2017
### LCKurtz
The set of all polynomials is too large to be a basis for $C_0$, but $\{1,x,x^2,\dots\}$ would do.
5. Jul 12, 2017
### mathwonk
this is very nice, and shows a creative grasp of what you are learning. keep it up and you will eventually do some new research! | {
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6. Jul 13, 2017
### Krylov
While I usually like the straightforward approach best (maybe this is one of the differences between a "pure" mathematician and an (aspiring) "applied" mathematician?), I gave it some thought and then came to the conclusion that I agree with you.
Just as an exercise, you could try to complete the direct approach as well. (You are almost there, anyway.)
7. Jul 13, 2017
### Rodrigo Schmidt
Hm, i see, but isn't that the basis of the set of all polynomials? So, by being a basis, that must be a linearly independent set, and it's also infinite, so that implies in the infinite dimensionality of C°(ℝ)?
That's where i want to get someday. Thanks for the inspiration!
8. Jul 13, 2017
### Krylov
I think that I and post #4 misunderstood you when you wrote
We (or I, at least) thought that here you asserted that the set of all polynomials itself is a basis for a linear subspace of $C^0(\mathbb{R})$, maybe because you wrote "the basis". However, from what you wrote afterwards (quoted at the top of this post), I get that you meant any (Hamel) basis for the linear space of all polyniomials. (For example, you can indeed use the canonical basis mentioned in post #4. By the way, I think there the $C_0$ was written by small mistake.)
Very good, as far as I can see, you are more than done.
Last edited: Jul 13, 2017
9. Jul 13, 2017
### Rodrigo Schmidt
Yeah, that's what i meant. Sorry if i wasn't clear enough! English is not my mother language so, mostly when writing about math and science, my texts can get a little confusing. I will try to be more specific in the next time.
Okay! Thanks for the support and knowledge shared! | {
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