# Tagged Questions

Use this tag only if your question is about the modern theoretical footing for probability, for example probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use [tag:probability-...

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### Upper bounds on $E[{\rm var}^2(X|Y)]$

I am looking for an upper bound on the quantity \begin{align*} E[{ \rm var}^2(X|Y)] \end{align*} where ${\rm var}(X|Y)=E[(X-E[X|Y])^2|Y]$. Getting a lower bound is rather easy using Jensen's ...
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### Computing a conditional expectation for uniform RVs

Suppose $X_1, ..., X_n \sim U[0, 1]$ are iid uniform RVs. How would I go about computing $E[X_n | X_{(n)}]$ where $X_{(n)}$ is the nth order statistic, i.e. $\max\{X_1, ..., X_n\}$ ? I'm stuck ...
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### Expected time to fill a table [duplicate]

Say I have a table of numbers 1-6. I throw a 6 sided die a number of times. Each time I get a number I have not already had, I mark it in the table. What is the expected number of times to throw the ...
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### Roll a die until, for the first time, the same side shows up two times in a row. Let $X$ be the number of rolls needed. compute $\mathbb{E}(X)$.

I'm having trouble with solving this problem: Roll a die until, for the first time, the same side shows up two times in a row. Let $X$ be the number of rolls needed. compute $\mathbb{E}(X)$. I know ...
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### Take $k$ shoes ($k \leqslant n$) from a wardrobe. What is the expected value of the number of pairs ($X$) you take?

I'm having trouble with this question: Let there be $2n$ shoes ($n$ pairs) in a wardrobe, arbitrary ordened. Take $k$ shoes ($k \leqslant n$) from that wardrobe. What is the expected value of the ...
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### Impossible Events, Probability Zero Events, Change of Sample Space, Invariant, Canonical Sample Space?

I am reading this post about probability theory and its foundations by T. Tao, and also this and this post, and they say in essence that the underlying sample space is not that much important. Often ...
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### If $P(a_n\leq X_n\leq b_n)\to1$ and $a_n\to0,b_n\to0$ then $X_n\to\delta_0$ in distribution

If $P(a_n\leq X_n\leq b_n)\to1$ and $a_n\to0,b_n\to0$ then prove that $X_n\to\delta_0$ in distribution. Here $\delta_0$ is the degenerate random variable putting all its mass at the point $0$. I do ...
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### Cumulative distribution function of Cauchy distribution

Let X be a Cauchy distribution with X~Cauchy(1) (so a=1). Prove that Y=1/X has the same cumulative distrubtion as X. Now I've tried taking F_X(x) for a=1 combined with the identity arctan(x)+...
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### time it takes to service a car with exponential random variable with rate 1

Need help with this question here. Ill post exactly what it says then show my ideas so far. "The time it takes to service a car is an exponential random variable with rate 1. (a) If A brings his car ...
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### Berry-Esseen bound for binomial distribution

From the Berry-Essen theorem I can deduce $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - \Phi(x)\right| \le \frac{C(p^2+q^2)}{\sqrt{npq}}$$ with $C \le 0.4748$. My ...
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### The length of the set which will be covered by Brownian motion in a time $t$

I have the following question in mind which I wanted to answer: what is the measure of the set which will be covered by a standard Brownian motion $B(t)$ in a time $t$? Call this random variable $M(t)$...
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### Law of large numbers for nonnegative random variables [closed]

I'm struggling with specific variation of Strong Law of Large Numbers. Suppose $X_1,X_2,\ldots$ are independent, identically distributed, nonnegative random variables and $\mathbb{E} X_1 = \infty$. ...
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### Show that $(W_t)_{t \geq 0}$ and $(W_t^2-t)_{t \geq 0}$ are not uniformly integrable

I'm considering the following martingale $M_t:=W_t^2-t,\ t\geq 1$, where the $W_t$ is a Brownian motion. I want to prove that this martingale and the Brownian motion are not uniformly integrable. I ...
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### Why is the supremum a random variable in the Glivenko–Cantelli theorem

According to wikipedia: Assume that $X_1,X_2,\dots$ are independent and identically-distributed random variables in $\mathbb{R}$ with common cumulative distribution function $F(x)$. The empirical ...
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### mutual information adds along path

Is it true that $I(X;Y)+I(Y;Z)=I(X;Z)$ for $X \to Y \to Z$? $I(X;Z) = H(X)+H(Z)-H(X,Z)$ and $I(X;Y)+I(Y;Z) = H(X)+H(Z)-H(Z|Y)-H(X|Y)$ Hence, we would require $-H(X,Z)=-H(Z|Y)-H(X|Y)$ -- is it true? ...
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### Computing $p(d|e_1,e_2)$ from $p(d|e_1)$ and $p(d|e_2)$

I know the probability $p(d|e_1)$ and $p(d|e_2)$, how to compute the $p(d|e_1, e_2)$ if $e_1$ and $e_2$ are independent? What if $e_1$ and $e_2$ is dependent, how to compute?
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### A convergent sequence of normal random variables

Say $\{X_n\}$ is a sequence of normal random variables with means $0$ and variances $\sigma_n^2$. Also suppose that $X_n\to X$ (everywhere) and $\sigma_n^2\to \sigma^2.$ Then, using characteristic ...
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### From pairwise P(A > B), to P(A > all distributions in set)

$\{D_0,…,D_n\}$ is a finite collection of independent (but not necessarily identically distributed) random variables. Define $f(x,y)=P(D_x≥D_y)$ and $g(x)=P(∀y:D_x≥D_y)$. Does $f$ determine $g$, and ...
I have a problem with this exercise. I completely do not know hot to tackle it. Please help. A bin contains $N$ strings. You randomly choose two loose ends and tie them up. You continue until there ...