# Tagged Questions

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### Show expectation is infinite

Let $X_1,\ldots,X_n$ be independent, identically distributed with expectation 1 and finite variance. Find the limit distribution of $\sqrt{n}(\bar{X}_n^{-1}-1)$. If the random variables are sampled ...
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### Conditional probability with balls in urns involving discards

I found this problem in a statistics book, and I'm wondering if my solution is correct. "You and a friend play a game involving 20 balls in an urn, of which 1 is red and 19 are white. The game is ...
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### Please check the question: Compute $EX$

Question: A box contains $10$ balls numbered $1,2,\ldots,10$. A random sample of $7$ balls is selected. $X=$ the smallest of the numbers drawn. Compute $E(X)$ $R(X)= \{1, 2, 3, 4\}$ ...
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### Probability question.

How many ways are there to distribute 2 indistinguishable white and 4 indistinquisable black balls into 4 indistinquisable boxes? If the question is asked as "distinct boxes", I can solve. But now, ...
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### Conditional probability question

Please check the conditional probabilty question I posted. I solved this. But I am not sure. Thank you:)
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### A conditional probabilty question.

Question: $8$ identical balls are randomly distributed into $8$ boxes. Given first box and second box are not both empty, find the probability that first box is not empty? $A:=$ B1 is not ...
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### Two probability questions.

I have two questions. (1) solution(1): Sample size $=|S|=12^{20}$ $11^{20}\rightarrow$ guarantee that one box is empty. $10^{20}\rightarrow$ guarantee that two boxes are empty. ...
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### Show that the intersection of two probabilities in a certain interval

I am struggeling with the following problem: Suppose that $P(A)= \frac{3}{4}$ and $P(B)= \frac{1}{3}$. Show that $\frac{1}{12} \leq P(A \cap B) \leq \frac{1}{3}$. Basically I try to show this ...
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### how many ways are there to distribute 10white and 10black balls into 20 distinct boxes so that at most one box is empty

Question: how many ways are there to distribute 10white and 10black balls into 20 distinct boxes so that at most one box is empty. I understand case-1. But I cannot understand a part of answer ...
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### Why is an algebra not a $\sigma$-algebra by induction?

I am studying probability theory by reading Sidney Resnick's "A Probability Path". On page 12 and 13, algebra and $\sigma$-algebra are defined. The only difference between the two is the third ...
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### Inverse Fourier Transform of $S_Y(f)$

I have this power spectral density $$S_Y(f) =\frac{N_0}{4 \pi ^{2} f^{2}}\left [ 1- \cos(2\pi f T) \right ]$$ Can any one help me how to find the Inverse Fourier transform?
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### What can I do with measure theory that I can't with probability and statistics

I've studied mathematics and statistics at undergraduate level and am pretty happy with the main concepts. However, I've come across measure theory several times, and I know it is a basis for ...
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### PMF of Two Random Variables

X and Y are independent and geometrically distributed random variables with $$P(X = m) = p(1-p)^{m}, m=0,1,2...$$ $$P(Y = n) = p(1-p)^{n}, n=0,1,2...$$ To find the probability mass function ...
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### Expected value of sample median given the sample mean.

Let $Y$ denote the median and let $\bar{X}$ denote the mean of a random sample of size $n=2k+1$ from a distribution that is $N(\mu,\sigma^2)$. How can I compute $E(Y|\bar{X}=\bar{x})$? Intuitively, ...
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### A question on conditional probability

Question: Let $X$ and $Y$ be two random variables. The relationship between the two is as follows. If $Y$ is less than or equal to $1$, then $X$ is equal to $Y$; If $Y$ is more than $1$, then $X$ ...
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### Examples of Talagrand's inequality

I am trying to understand Talagrand's inequality and when it gives better results than Markov/Chebyshev/Chernoff. However I find the formal definition hard to understand. Are there any nice simple ...
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### Shooting Star (Probability)

Assume that a random experiment consists in centering a telescopic sight on a random star. Let $A_{n}$ ($n \in \mathbb{Z}^{+}$) denote the event that the telescopic sight spots exactly $n$ stars. ...
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### Square Root of Random Variables

Question: Suppose that $\displaystyle \frac{2}{\theta_0}\sum_{i=1}^n y_i\sim\displaystyle\chi_{2n}^2$ and $\displaystyle 2\theta_0\sum_{i=1}^n x_i\sim\displaystyle\chi_{2n}^2$. And these two are ...