text stringlengths 1 2.12k | source dict |
|---|---|
Test Cases
1. $$[2, 3, 5, 6, 6.6, 6.7, 7, 10]$$, $$\text{target} = 9$$
2. $$[2, 3, 5, 6, 6.6, 6.7, 7, 10]$$, $$\text{target} = 6.25$$
3. $$[2, 3, 5, 6, 6.6, 6.7, 7, 10]$$, $$\text{target} = 6.7$$
4. $$[2, 3, 5, 6, 6.6, 6.7, 7, 10]$$, $$\text{target} = 6.1$$
5. $$[2, 3, 5, 6, 6.6, 6.7, 7, 10]$$, $$\text{target} = 3.27$... | {
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# The nth root lies between the 1, and the number
start: int = 1
end: int = number
while (end - start) > THRESHOLD:
middle: float = (start + end) / 2.0
if get_N_power(middle, root) > number:
end = middle
elif get_N_power(middle, root) < number:
start = middle
else:
return middle
return start, end
find_n_th_root(1024... | {
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### Apporach
#### Naive Approach
• Iterate over all the elements, and then sort them,
• then return the middle element.
#### Time Complexity for this naive apporach
• $$O(NM)$$ for the traversal,
• $$O(NM \log MN)$$ for Sorting and,
• Constant $$O(1)$$ time for the middle element. So total of $$O(NM \log MN)$$.
##... | {
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# Calculating number of equivalence classes where two points are equivalent if they can be joined by a continuous path.
Q. Let $G$ be an open set in $\Bbb R^n$. Two points $x,y \in G$ are said to be equivalent if they can be joined by a continuous path completely lying inside $G$. Number of equivalence classes is
1. ... | {
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It is impossible to have uncountably many equivalence classes. Note that each equivalence class is an open set, since balls are path-connected and so if $x\in G$ then any open ball around $x$ contained in $G$ is in the same equivalence class. Now any nonempty open subset of $\mathbb{R}^n$ contains an element of $\mathb... | {
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# Clausen sum
#### ZaidAlyafey
##### Well-known member
MHB Math Helper
Evaluate the following
$$\displaystyle \sum_{k\geq 1}\frac{\cos \left(\frac{\pi k}{2} \right)}{k^2}$$
#### M R
##### Active member
It looks like part of the Fourier series for $$\displaystyle x(\pi-x)$$ at $$\displaystyle x=\pi/4$$
So I'm get... | {
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... |
$$\displaystyle \frac{\pi i\theta}{2}-\frac{i\theta^2}{4}-\int_0^{\theta}\log(1-e^{-ix})\,dx$$
Now we expand the complex logarithm within the integrand as a power series:
$$\displaystyle \log(1-z)=-\sum_{k=0}^{\infty}\frac{z^k}{k}$$
$$\displaystyle \Rightarrow$$
$$\displaystyle \int_0^{\theta}\log(1-e^{-ix})\,dx=-\... | {
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... |
##### Well-known member
Thanks Z!
It can make tricky-looking results quite easy. Such as
$$\displaystyle \sum_{k=1}^{\infty}\frac{\cos (\pi k/10)}{k^2}=\frac{\pi^2}{6}-\frac{\pi}{4}\sqrt{\frac{5+\sqrt{5}}{2}}+\frac{5+ \sqrt{5}}{16}$$
#### DreamWeaver
##### Well-known member
Thanks Z!
It can make tricky-looking res... | {
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... |
# Integral domains with non-trivial group of units that are not fields
I'm looking for examples of integral domains that are not fields but at the same time have more units than just the multiplicative identity 1.
It's clear to me that by Wedderburn's little theorem, there are no finite examples of this type.
If I u... | {
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# Is there a way I can find out the degree of this extension without explicitly finding the minimal polynomial?
Suppose that $\beta$ is a real cube root of $2$ and $\omega$ is a primitive third root of unity. I've a problem that asks me to show that if $\alpha=\beta\omega$, then $\alpha+\beta$ has minimal polynomial o... | {
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• Good answer! +1 – Learnmore Nov 23 '16 at 7:09
• Excellent strategy in the second part. I've been trying to prove the irreducibility of $x^6+108$ for some time. This will help in similar other problems too I think, so thank you. – adrija Nov 23 '16 at 7:32
• Thanks, glad to be of help. – carmichael561 Nov 23 '16 at 1... | {
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And this list is exhaustive. The minimal polynomial of $\sqrt[3]{2} + \omega \sqrt[3]{2}$ over $\mathbb{Q}$ therefore has $3$ roots and is of degree $3$. Repeating this process for $- \sqrt[3]{2} + \omega \sqrt[3]{2}$ will demonstrate that its minimal polynomial is of degree $6$.
At least for degree 3, it's not too ha... | {
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# Why is mean ± 2*SEM (95% confidence interval) overlapping, but the p-value is 0.05?
I have data as two lists:
acol = [8.48, 9.82, 9.66, 9.81, 9.23, 10.35, 10.08, 11.05, 8.63, 9.52, 10.88, 10.05, 10.45, 10.0, 9.97, 12.02, 11.48, 9.53, 9.98, 10.69, 10.29, 9.74, 8.92, 11.94, 9.04, 11.42, 8.88, 10.62, 9.38, 12.56, 10.5... | {
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A summary of the above lists is given below:
N, Mean, SD, SEM, 95% CIs
137 9.92 1.08 0.092 (9.74, 10.1)
137 10.2 0.951 0.081 (10.0, 10.3)
An unpaired t-test for the above data gives a p-value of 0.05:
f,p = scipy.stats.ttest_ind(acol, bcol)
print(f, p)
-1.9644209241736 0.050499295018989004
I understa... | {
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So the following are true for independent $$\bar{X}_1$$ and $$\bar{X}_2$$:
$$\begin{array}{} \text{Var}(\bar{X}_1-\bar{X}_2) &=& \text{Var}(\bar{X}_1) + \text{Var}(\bar{X}_2)\\ \sigma_{\bar{X}_1-\bar{X}_2}^2 &=& \sigma_{\bar{X}_1}^2+\sigma_{\bar{X}_2}^2\\ \sigma_{\bar{X}_1-\bar{X}_2} &=& \sqrt{\sigma_{\bar{X}_1}^2+\si... | {
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• Instead of z-values and a z-test you are actually doing (should be doing) a t-test. So it might be that the levels on which you base the confidence intervals for the error bars (like '95% is equivalent to 2 times the standard error') will be different for the t-test. To be fair, to compare apples with apples, you sho... | {
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• It will be great if you could explain your long sentence / short para on s1+s2, its double, distance and 2*SEM. That will help us understand this concept better. – rnso Nov 21 '20 at 17:25
• I expected standard deviation (even pooled) to be much greater than (SEM1+SEM2). I tried with random data and find that it is a... | {
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If it is only the whiskers that overlap or touch, then the null hypothesis will produce this result a lot less often than 5%. This is because (to use your example) both the blue sample would need to be low, and at the same time the red sample would need to be high (exactly how high would depend on the blue value). You ... | {
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When you look at the overlap, you are seeing points for which the difference is less than two standard deviations. But remember that the variance of the difference between two variables is the sum of the individual variances. So you can generally use a rule of thumb that if you want to compare two different populations... | {
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# Charge to mass ratio inversely proportional to curved path radius?
In a cloud or bubble chamber, charged particles follow circular paths. I learned that charge to mass ratio of the particles is inversely proportional to the radius of the path. Thus, a particle following a circular path with a large radius means that... | {
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• is B/v always going to be a constant. I mean, can I use this relationship to compare the curved paths of two different particles (resulting from the breakdown of one particle in a bubble chamber)? Dec 7 '13 at 3:53
It boils down to balancing the centripetal force, $$\vec{F}=\frac{mv^2}{r}\hat{r}$$ with the magnetic ... | {
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# The set of all limit points of a set
Given a set $A \subseteq \mathbf{R}$, let $L$ be the set of all limit points of $A$. I recently worked through the proof that $L$ is closed. However, I have a few queries thinking deeper about this result.
Since $L$ is closed, this implies that if $x$ is a limit point of $L$ the... | {
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Consider. If $$A = [0,1]$$ then $$L = [0,1] = A$$ and set of limit points of $$L = [0,1] = A$$. But $$[0,1]$$ is not a limit point of $$A$$! Contradiction? No. A set is not a member of itself (by ZFC that can never happen) so the set of limit points is not a limit point.
Clearly, L={0} but does not have any limit poin... | {
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Therefore, this set of ordered pairs comprises of n, pairs. Don’t stop learning now. • Encode R Encode R Let A = {1, 2, 3}. [1] [2] An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself. Confirm that R is a reflexive relation on set A. No... | {
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matrix. Now, p can be chosen in n number of ways and so can q. Anti - Reflexive: If the elements of the set do not relate to themselves, they are said to be irreflexive or anti-reflexive. If A = {1,2,3} the number of reflexive relations in 'A' are 1 See answer radhasri8306 is waiting for your help. An example of a refl... | {
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How to swap two numbers without using a temporary variable? acknowledge that you have read and understood our, GATE CS Original Papers and Official Keys, ISRO CS Original Papers and Official Keys, ISRO CS Syllabus for Scientist/Engineer Exam, For every set bit of a number toggle bits of other, Toggle bits of a number e... | {
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is impossible for a reflexive $\Leftrightarrow$ (,. More fundamental and rigid framework for these concepts report any issue with properties! That all positive integers are included in these ordered pairs comprises of n pairs of such ( p, )! Now to bookmark this formula work for number of reflexive relations from set a... | {
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above.! Also, there will be a total of n pairs of such ( p, q ) and! To bookmark abstract concepts in math, like infinity in set S linked! At contribute @ geeksforgeeks.org to report any issue with the above content if the matrix total number of reflexive is! X > y ) on the GeeksforGeeks main page and help other Geeks ... | {
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## Welcome to Mathematics Stack Exchange
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's built and run by you as part of the Stack Exchange network of Q&A sites. With your help, we're working together to build a library of detaile... | {
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Use edits to fix mistakes, improve formatting, or clarify the meaning of a post.
You can always comment on your own questions and answers. Once... | {
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# Continuity of an inverse function
## Homework Statement
Prove that the a continuous function with compact domain has a continuous inverse. Also prove that the result does not hold if the domain is not compact.
## The Attempt at a Solution
I tried using the epsilon delta definition of continuity but didn't get any... | {
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Dick
Homework Helper
oh sorry, i also forgot to add that
f:[a,b] => R
I got the continuity part down (your hint really helped!), but I'm having trouble with the compact part.
What kind of 'trouble with the compact part'? You are given that the domain is compact, you don't have to prove it.
I'm having trouble with givi... | {
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# 6 women, 7 men form a pair of 1 woman 1 man. How many ways can we select 3 pairs.
So the way to do this I think is to 6C3*7C3*9 (where 9 is the number of ways we can arrange the men and women within the 3 pairs).
But I can't seem to figure out how I would do this in a different approach where I look at the total pa... | {
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Method 2: We choose the women, then match the men to the women.
There are $\binom{6}{3}$ ways of selecting three of the six women. We line them up in some order. We can match the first woman in the line to one of the seven men, the second woman in the line to one of the six remaining men, and the third woman in the li... | {
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We have $30$ pairs from which to choose the second pair. However, once we choose a pair, we need to get rid of $4$ other pairs with the same man (since there are $6-2=4$ women who haven't been picked yet) and the $5$ other pairs with the same woman (since there are $7-2=5$ men who haven't picked yet), leaving us with $... | {
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# System of two Equations
A friend of Mine gave me a system of two equations and asked me to solve them $\rightarrow$
$$\sqrt{x}+y=11~~ ...1$$ $$\sqrt{y}+x=7~~ ...2$$
I tried to solve them manually and got this horrendously complicated fourth degree equation $\rightarrow$
\begin{align*} y &= (7-x)^2 ~...\mbox{(from... | {
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P.S. I'm In high-school.
-
From the fourth degree equation, you can use Ferrari's method (en.wikipedia.org/wiki/Quartic_function) to solve it. – guaraqe Jun 1 '12 at 15:05
In this question a solution for the same system (with $x,y$ interchanged) is asked. As such the present question is a duplicate. $$\sqrt{x} + y = ... | {
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Assume $x$ and $y$ are integers. Notice that, in this case, if $\sqrt x +y=11$, an integer, then $\sqrt x$ must be an integer. A similar argument can be made for $y$. So if they're integers then they're both perfect squares. Rephrasing in terms of the square roots (still integers) $X=\sqrt x,Y=\sqrt y$ $$X+Y^2=11$$ $$Y... | {
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Once you guessed the solutions, you can easily prove that there are no others. Rewrite the equations as $y=11-\sqrt x=F(x)$ and $x=7-\sqrt y=G(y)$. Note that both $x,y\le 11$, so their square roots are at most $4$, which means that $x,y\ge 3$. Now just observe that $z\mapsto \sqrt z$ is a contraction on $[3,\infty)$ (t... | {
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This notebook contains an excerpt from the Python Programming and Numerical Methods - A Guide for Engineers and Scientists, the content is also available at Berkeley Python Numerical Methods.
The copyright of the book belongs to Elsevier. We also have this interactive book online for a better learning experience. The ... | {
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P, L, U = lu(A)
print('P:\n', P)
print('L:\n', L)
print('U:\n', U)
print('LU:\n',np.dot(L, U))
P:
[[0. 0. 1.]
[0. 1. 0.]
[1. 0. 0.]]
L:
[[ 1. 0. 0. ]
[-0.25 1. 0. ]
[ 0.5 0.5 1. ]]
U:
[[ 8. 8. 0. ]
[ 0. -2. 5. ]
[ 0. 0. -7.5]]
LU:
[[ 8. 8. 0.]
[-2. -4. 5.]
[ 4. 3. -5.]]
We can see the ... | {
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# Gauss's Law Problem
## Homework Statement
So here's the question...
A cylindrical shell of radius 7.00 cm and length 240 cm has its charge uniformly distributed on its curved surface. The magnitude of the electric field at a point 19.0 cm radially outward from its axis (mesaured from the midpoint of the shell) is ... | {
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Line charge:
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elecyl.html#c1
conducting cylinder:
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elecyl.html#c2
So, just to make sure I'm on the right page here, we can basically consider the conducting cylinder as a really thick line charge?
LowlyPion
Homework ... | {
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# Evaluating a limit by integral test
1. Apr 20, 2014
### MathewsMD
1. The problem statement, all variables and given/known data
Evaluate $∑^∞_{n=1} \frac {2}{n(n+2)}$
2. The attempt at a solution
I've solved this question simply enough by evaluating it as a telescoping series and found the answer as 3/2. Now, wh... | {
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### MathewsMD
In general? Do you mind expanding please? :)
7. Apr 20, 2014
### Staff: Mentor
I didn't want to say a flat no, just in case there was some situation that I hadn't thought about. The important thing is that the integral test is just a test to determine whether a given series converges or not.
8. Apr 2... | {
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IntMath Home » Forum home » Methods of Integration » Find integral sqrt (x^2 + 1) using trigonometric substitution
# Find integral sqrt (x^2 + 1) using trigonometric substitution [Solved!]
### My question
Find int sqrt (x^2 + 1) dx with limits of integration from 0 to 1 using Trigonometric Substitution.
### Relevan... | {
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When we change x to tan theta and x goes from 0 to 1, what will theta's lower and upper values be?
X
@phinah: Good on you for using the math entry system! (I tidied up some of the math expressions.)
Just a small (but important) point - don't miss out the "d theta" parts in your second line. It should be:
Replacing ... | {
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## Re: Find integral sqrt (x^2 + 1) using trigonometric substitution
Leaving it in terms of theta:
int sec^3 theta d theta from 0 to .7853
According to Wolfram, the integral formula is
1/2[tan theta sec theta {: - ln (cos {:theta/2:} - sin {:theta /2:}) {: + ln (sin {:theta/2:} + cos {:theta/2:})]
Therefore, 1/2 [... | {
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## Re: Find integral sqrt (x^2 + 1) using trigonometric substitution
Got it. Thanks.
After finding the integral in terms of theta you state that we can change back to x or leave it in terms of theta. Above, I left it in terms of theta.
Part Two is to change it back to x. Trying to figure out why I did not arrive at ... | {
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## Re: Find integral sqrt (x^2 + 1) using trigonometric substitution
But we can't just do this (trade thetas for xs)!
int sec^3 x d x
=1/2[tan x sec x - ln (cos (x/2) - sin (x/2)) {: + ln (sin (x/2) + cos (x/2))]
The expressions in theta need to stay in theta, then we substitute back into x. You kind of did bits of... | {
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cos(theta/2) = sqrt((1 + 1/(sqrt(x^2+1)))/2)
Do you think you can proceed from there?
## Re: Find integral sqrt (x^2 + 1) using trigonometric substitution
The half-angle formula for sine is similar to that for cosine only we subtract in the numerator.
Substituting the limits of integration:
1/2 [1( sqrt 2) - ln(.9... | {
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Question
# Let $$A$$ be the sum of the first $$20$$ terms and $$B$$ be the sum of the first $$40$$ terms of the series $$1^{2} + 2.2^{2} + 3^{2} + 2.4^{2} + 5^{2} + 2.6^{2} + .....$$ If $$B - 2A = 100\lambda$$, then $$\lambda$$ is equal to
A
464
B
496
C
232
D
248
Solution
## The correct option is C $$248$$$$B=1^2+2... | {
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# A nice way to remember trigonometric integrals?
Is there a "nice" way to remember trigonometric integrals, beyond what is typically taught in a standard calculus class? I'm currently in Calculus II, and up to now I've found calculus rather accessible. I love that, at least in my classes, we learn the "how" and the "... | {
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Similarly, when you see $$1-x^2$$ you should be thinking $$x=\sin\theta$$ or $$x = \cos \theta$$ and the expression becomes $$1-\sin^2\theta = \cos^2\theta$$
And when you see $$x^2 - 1$$ it is a bit of a toss up. Sometimes, $$x = \sin \theta$$ works and sometimes $$x = \sec\theta$$ works better. It really has to do wh... | {
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One more example
$$\int \frac {1}{x^2+x+1} \ dx$$
The denominator looks like a bit of a bear. It doesn't factor, if it did, I would suggest partial fractions. As it doesn't we use "completing the square."
$$x^2 + x + 1 = (x+\frac 12)^2 + \frac 34$$
$$\int \frac {1}{(x+\frac 12)^2 + \frac 34} \ dx$$
$$x+\frac 12 = ... | {
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# probability of obtaining at least one 6 if it is known that all three dice showed different faces [closed]
Three dice are rolled. What is the probability of obtaining at least one 6 if it is known that all three dice showed different faces? The answer is 0.5. Could you give a hint?
• I calculated the probability of... | {
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3)All combinations of different numbers are equally likely. $3$ numbers appear. $3$ do not. Each is equally likely so that a six appears (or not) is 1/2.
To justify the answer formally you could use what you know about conditional probability. In particular:
Let $S$ be the event that at least one six occurs, and $D$ ... | {
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• But can I calculate it using a formula? – Mary Oct 22 '16 at 22:43
• One way would be $\frac{\text{# of ways to pick 3 numbers with one being a 6}}{\text{# of ways to pick 3 numbers}}$ which would be $\frac{5\choose2}{6\choose3}$ = $\frac{1}{2}$ – turkeyhundt Oct 22 '16 at 22:50 | {
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# Where do we need absolute values when solving $u'(t) = \frac{u^2(t) - u(t)}{t}$
Solve the differential equation $$u'(t) = \frac{u^2(t) - u(t)}{t}$$ and then solve the initial value problem with $$u(1) = \frac{1}{2}$$.
I know the general solution is given by $$u(t) = \frac{1}{c_1 t + 1}$$ for all $$t \in \mathbb{R}$... | {
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• You can say $u/(u-1)=Ct$ where $C$ is a constant. If, instead, $C$'s sign changed with $t$, it would have to change only when $t=0$ in order to retain continuity, and then I suspect it would need to not change at all to be differentiable at $t=0$. – runway44 Jul 27 at 2:07
• That was your original last equation witho... | {
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Then you continue with (with the correct signs in the partial fraction decomposition) \begin{align} \frac{u(t_0)(u(t)-1)}{u(t)(u(t_0)-1)}&=\frac{t}{t_0}\\[.8em] u(t)\Bigl(t_0u(t_0)-t(u(t_0)-1)\Bigr)&=t_0u(t_0)\\[.8em] u(t)=\frac{t_0u(t_0)}{t_0u(t_0)+(1-u(t_0))t} \end{align}
• For your second part: The first task is to... | {
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or integrating
$$\frac 1v = \frac 1t + C\Rightarrow v = \frac{t}{C t+1} = u t$$
• As detailed in the other answer my approach is also correct and mandates the use of absolute values. By a uniqueness argument we can conclude that we can ignore these absolute values. Do you use that argument implicitly somewhere? I fin... | {
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The maximal $$(t,u)$$-domain relevant to the given IVP is $$\Omega:={\mathbb R}_{>0}\times{\mathbb R}$$. Within $$\Omega$$ the standard existence and uniqueness theorem for ODEs is valid. By inspection one sees that there are the constant solutions $$u_0(t)\equiv0$$ and $$u_1(t)\equiv1$$ $$(t>0)$$. No other solution ca... | {
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# Math Help - how to compute this probability?
1. ## how to compute this probability?
I have taken this problem from the paper of MASTER OF COMMERCE Examination of university in India with little modifications.
If a machine is correctly set up it will produce 90% acceptable items. If it is incorrectly set up it will ... | {
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Tell us how you approached this and then we might be able to see where you went wrong.
CB
Firstly, I am giving calculations for the second answer
Let us define the events:
$A_1=$The set up was correct.
$A_2=$The set up was wrong.
E=The item is acceptable.
D=The item is not acceptable (defective)
We know $P(A_1)=$Prob... | {
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$P(\text{3 acceptable}|\text{setup OK})P(\text{setup OK})=0.9^3\times 0.8$
and the denominator:
$P(\text{3 acceptable}|\text{setup OK})P(\text{setup OK})+P(\text{3 acceptable}|\text{setup NOT OK})P(\text{setup NOT OK})\\ \phantom{xxxxxxxx}=0.9^3\times 0.8+0.4^3\times 0.2$
CB
5. ## Re: how to compute this probabilit... | {
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# Valid AM-GM inequality proof?
We have to prove that $$\frac{(x_1+x_2+x_3+...+x_n)}{n} \geq (x_1\cdot x_2\cdot x_3\cdots x_n)^{1/n}$$
Attempt: Raising both sides to the nth power gives $\left(\frac{x_1+x_2+x_3+...x_n}{n}\right)^{n} \geq x_1x_2x_3...x_n$ This is equivalent to proving that if a sum of random numbers i... | {
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• meta.math.stackexchange.com/questions/5020/… – Timbuc Jun 21 '15 at 9:59
• I fixed the LaTeX (math notation). Please look to my edit and to the tutorial in the link of Timbuc's comment, to learn it. – wythagoras Jun 21 '15 at 10:01
• It seems mathematically correct to me, though I would expand on why (x1+x2...xn)^n i... | {
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• That's really interesting, I actually thought about the continuity of the process,(Replacing 2 numbers in the set with their arithmetic mean), each time you do this process you increase the product,and since the product is bounded, then, this process must "stop" somewhere which means that it must converge, and if it ... | {
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• And the downvote is because...? – Aaron Maroja Jun 21 '15 at 14:24
• Other users than the downvoter themself can only speculate what was the reason for the downvote. But if you wish to discuss the reason for the downvote, there is a chatroom explicitly for this purpose. – Martin Sleziak Jun 21 '15 at 16:44
• The prob... | {
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# Conditional Probability with coins (Edited with Progress)
You have two coins that look identical, but one of them is fair and the other is weighted. The weighted coin has a 3/4 probability to flip heads and a 1/4 probability to flip tails. You forget which coin is which, so you test each coin by flipping it once. If... | {
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$G$: guess correctly
$P(G|X)=\frac{1}{2}$ since if both come up same then random pick.
$$P(G|X^C) = \frac{P(FT \cap UH)}{P(X)}=\frac{\frac{1}{2}*\frac{3}{4}}{\frac{1}{2}}=\frac{3}{4}$$
So:
$$P(G) = P(G \cap X) +P(G \cap X^C)= P(X)*P(G|X)+P(X^C)*P(G|X^C)=$$
$$\frac{1}{2}*\frac{1}{2}+\frac{1}{2}*\frac{3}{4}=\frac{5}... | {
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# Probability that warehouse falls behind schedule
A warehouse uses three machines ($$m_1$$, $$m_2$$ and $$m_3$$) and their failure rate is 0.02, 0.03 and 0.04.
a) Find the probability that two or more machines fail.
b) The warehouse falls behind schedule if at least one of the following happens: (1) $$m_1$$ fails; ... | {
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• Your logic for $a$ appears to be incorrect. the event "$m_1$ and $m_2$ fail " (for one) is contained in the event "all three fail" so you are badly overcounting the latter scenario.
– lulu
Feb 8, 2018 at 20:03
• So would it just be: $P($two or more machines fail $) = P(m_1$ and $m_2$ fail$) + P(m_1$ and $m_3$ fail$)+... | {
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If $m_3$ fails, the warehouse falls behind schedule if at least one of the other machines fails.
Assuming independence, the desired probability is \begin{align*} P(F_1 \cup F_2) & = P(F_1) + P(F_2) - P(F_1 \cap F_2)\\ & = 0.02 + 0.03 - (0.02)(0.03)\\ & = 0.02 + 0.03 - 0.0006\\ & = 0.0494 \end{align*} More formally, th... | {
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# Relevance of prime being divisble by $4k+1$ in proof that 'There are infinitely many primes of the shape $4k+3$'
Show that there are infinitely many primes of the shape $4k+3$
Proof:
$1)$ Suppose that there are only finitely many such primes, say $p_1,...p_n$.
$2)$ Consider the integer $Q=4p_1...p_n-1$
$3)$ The ... | {
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Since you are looking for a new prime of form $4k+3$, you need to make sure that you have a candidate amongst the factors. If all the factors were of the form $4k+1$ you would have some larger primes than you had before, but they wouldn't be of the right form, and the proof would fail.
Fortunately it is east to show th... | {
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# Probability of Impurity Given 3 out of 5 test detections
A chemist is interested in determining whether a certain trace impurity is present in a product. An experiment has a probability of 0.80 of detecting the impurity if it is present. The probability of not detecting the impurity if it is absent in 0.90. The prob... | {
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The probability of not having an impurity present and getting three detections in five experiments is
$$\text{P}(P \,{\small\text{ AND }} \, 3 {\small\text{ out of }}5)=\left(\frac{3}{5}\right) \cdot \binom{5}{3} \left(\frac{1}{10}\right)^3\left(\frac{9}{10}\right)^2 = \frac{243}{50000}$$
We are looking for $$\,\text... | {
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• Thanks for the clarification!! Can I ask, how am I from that wording know the difference between conditional and intersection? It seems to me like this is the sensitivity of a test which is conditional? Nov 7, 2019 at 18:22
• This IS a conditional probability but it requires some intersection calculations first. The ... | {
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$$P(I|T^-)$$ is wrong.
$$P(I|T^-)=\frac{P(I\cap T^-)}{P(T^-)}=\frac{0.4\cdot 0.2}{0.4\cdot 0.2+0.6\cdot 0.9}=\frac{4}{31}$$
and you can guess from here that the subsequent use of binomial distribution is wrong too - you can't use it this way.
• Thanks for pointing this out! I was thinking $1 = P(A|B) + P(A|B^C)$ ins... | {
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1. ## More Probability
I never took Stats before, so I'm just trying to get a grasp of statistics and make sure I understand all of it.
A box in a certain supply room contains four 40-W lightbulbs, five 60-W bulbs, and six 75-W bulbs. Suppose that three bulbs are randomly selected.
(c) What is the probability that o... | {
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etc.
3. Hello, hansel13!
For part (d), I had to crank out an exhaustive list.
A box contains four 40w lightbulbs, five 60w bulbs, and six 75w bulbs.
(d) Suppose now that bulbs are to be selected one by one until a 75w bulb is found.
What is the probability that it is necessary to examine at least six bulbs?
There a... | {
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I got .0503496 from mr fantastic.
and .04195804 from Soroban.
Still a little confused as well.
Wouldn't the following just work?:
$
P = {{\binom{5}{0}}{\binom{9}{5}}}/{\binom{15}{5}} = .042
$
5. Originally Posted by hansel13
Still a little confused as well.
Wouldn't the following just work?:
$
P = {{\binom{5}{0}}{\... | {
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# Definition of the outer measure and the outer measure of an interval.
I'm taking a course in Real Analysis covering Measure Theory. We are using Real Analysis by Royden/Fitzpatrick as the text. I'm going through the book and I'm struggling to understand the approach of Proposition 1 which states: The outer measure o... | {
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• You have to prove that $m^*([a,b]) \geq b-a$. The above doesn't prove it. Sep 16, 2020 at 16:25
By definition of the infimum: the infimum of a subset $$S$$ of a partially ordered set $$T$$, denoted $$\inf S$$ is the greatest element in $$T$$ that is less than or equal to all elements of $$S$$.
Applying that, you hav... | {
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# Difference between $\lim P[…]$ and $P[ \lim ]$
In a Galton-Watson branching process the extinction probability is sometimes given by $$\lim_{t \rightarrow \infty} P[X(t)=0]$$ and sometimes as $$P[\lim_{t \rightarrow \infty}X(t)=0]$$ Is there a difference between These two formulations? Which is the "correct" one, if... | {
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• thank you for your answer. that helps a lot. Do you know a link to a paper, where the mathematical Definition of a Galton-Watson process is stated? All I can find is Wikipedia and the references there do not lead to formal definitions, either. – user146358 Aug 16 '14 at 12:57
• The WP page has a formal definition. So... | {
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# Modular arithmetic for arbitrary number
The problem describes as:
When the even integer $n$ is divided by $7$，the remainder is $3$. What's is The remainder when $n$ is divided by $14$.
My simple solution is:
$n=7x+3$ where $x$ is odd, so, we can define $x = 2m+1$, then $n = 7(2m+1) + 3 = 14m + 7 +3= 14m + 10$. So... | {
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$$n=14m+r$$
Then, let us suppose three possibilities:
1. $r \lt 7$
2. $r = 7$
3. $7 \lt r \lt 14$ so we can define $r=7+r'$, where $r' \lt 7$
• For the first case:
$$n=14m+r \pmod{7} \equiv 2\cdot 7m\pmod{7} + r \pmod{7} \equiv 0+r \equiv 3$$
Thus:
$$r=3$$
But that is not possible because it means that $n = 14... | {
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# Is is possible to find a basis for the column space of $A$,given reduced row echelon form of matrix $A$ and $A^T$,
Suppose $A$ is a $3$x$4$ matrix and the reduced row echelon form of $A$ is $\begin{pmatrix}1&0&0&1\\0&1&2&2&\\0&0&0&0\end{pmatrix}$
and the reduced row echelon form of $A^T$ is $\begin{pmatrix}1&0&2\\0... | {
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So instead of performing ECOs on $A$, we perform EROs on $A^T$.
This gives us row vectors (apart from the zero row vectors) that are linearly independent and span the row space of $A^T$, which is equivalent to the column space of $A$.
• do you agree with the commenter? – Anonymous Oct 26 '16 at 4:11
• @Anonymous Not ... | {
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# The colon does not align with the \vdots
I have the following code to align several equations:
\begin{aligned}
s_{1} & :=\sum_{1 \leq j \leq m} X_{j} \\
s_{2} & :=\sum_{1 \leq j<k \leq m} X_{j} \cdot X_{k} \\
& \vdots \\
s_{k} & :=\sum_{1 \leq j_{1}<j_{2}<\cdots<j_{k} \leq m} X_{j_{1}} \cdot X_{j_{2}} \cdot \cdots ... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9399133548753619,
"lm_q1q2_score": 0.8437557936604345,
"lm_q2_score": 0.897695292107347,
"openwebmath_perplexity": 4338.49176923962,
"openwebmath_score": 1.0000100135803223,
"tags"... |
\begin{align*}
s_{1} &\coloneqq\sum_{1 \leq j \leq m} X_{j} \\
s_{2} &\coloneqq\sum_{1 \leq j<k \leq m} X_{j} \cdot X_{k} \\
&\vdotswithin{\coloneqq} \\
s_{k} &\coloneqq\sum_{1 \leq j_{1}<j_{2}<\cdots<j_{k} \leq m}
X_{j_{1}} \cdot X_{j_{2}} \cdot \cdots \cdot X_{j_{k}} \\
&\vdotswithin{\coloneqq} \\
s_{m} &\coloneqq X_... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9399133548753619,
"lm_q1q2_score": 0.8437557936604345,
"lm_q2_score": 0.897695292107347,
"openwebmath_perplexity": 4338.49176923962,
"openwebmath_score": 1.0000100135803223,
"tags"... |
# Find $f(x)$ from $f(3x + 1)$
The problem that I have to solve is:
If the following function is valid for every value of $x$
$$f(3x + 1) = 9x^2 + 3x$$
find the function $f(x)$ and prove that for every $x\in\mathbb R$ the following is valid: $$f(2x) - 4f(x) = 2x$$
-
Here $$f(3x+1)=3x(3x+1)=((3x+1)-1)(3x+1)$$ $$\i... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9793540722737478,
"lm_q1q2_score": 0.8437509573074417,
"lm_q2_score": 0.8615382129861583,
"openwebmath_perplexity": 599.7775307576528,
"openwebmath_score": 0.8559978604316711,
"tag... |
-
Why would this be downvoted? – Pedro Tamaroff Aug 9 '12 at 19:12
@Downvoter If something is not clear then please feel welcome to ask for an explanation. – Bill Dubuque Aug 9 '12 at 19:12
I like this, but it took me quite a while to grasp! (Of course, I'm rather low on the "totem pole"...) – The Chaz 2.0 Aug 9 '12 at... | {
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"lm_q1_score": 0.9793540722737478,
"lm_q1q2_score": 0.8437509573074417,
"lm_q2_score": 0.8615382129861583,
"openwebmath_perplexity": 599.7775307576528,
"openwebmath_score": 0.8559978604316711,
"tag... |
# For what values of $x$ in $(-3,17)$ does the series $\sum\limits^{\infty}_{n=1}\frac{(-1)^n x^n}{n[\log (n+1)]^2}$ converge?
For what values of $x$ in the following series, does the series converge?
\begin{align}\sum^{\infty}_{n=1}\dfrac{(-1)^n x^n}{n[\log (n+1)]^2},\;\;-3<x<17 \end{align}
MY TRIAL
\begin{align}\... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9793540716711546,
"lm_q1q2_score": 0.8437509567882846,
"lm_q2_score": 0.8615382129861583,
"openwebmath_perplexity": 927.8465566898402,
"openwebmath_score": 0.9395433068275452,
"tag... |
You are correct. This is a "variation on the theme".
For $|x|>1$ $$\lim_{n\to +\infty}\dfrac{|-x|^n}{n[\log (n+1)]^2}=+\infty$$ and the series is divergent.
For $|x|\leq 1$, by direct comparison, the series is absolutely convergent $$\sum^{\infty}_{n=1}\dfrac{|-x|^n}{n[\log (n+1)]^2}\leq \sum^{\infty}_{n=1}\dfrac{1}{... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9793540716711546,
"lm_q1q2_score": 0.8437509567882846,
"lm_q2_score": 0.8615382129861583,
"openwebmath_perplexity": 927.8465566898402,
"openwebmath_score": 0.9395433068275452,
"tag... |
# How to find the launch angle for a ball from a fixed point where it is known the maximum height?
#### Chemist116
The problem is as follows:
From point indicated in the picture, a football player is about to kick a ball giving the ball a velocity of $v_{o}$. The projectile collisions with the crossbar on point $A$ ... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9793540674530016,
"lm_q1q2_score": 0.8437509531541845,
"lm_q2_score": 0.8615382129861583,
"openwebmath_perplexity": 388.41210262080216,
"openwebmath_score": 0.9000709056854248,
"tags"... |
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