# 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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### Let $X$ be normal with zero mean and variance $\sigma^2$, let $Y$ be uniform on $(0,\pi)$. Find the density of $Z=X+a \cos(Y)$.

Let $X$ be normal with zero mean and variance $\sigma^2$, let $Y$ be uniform on $(0,\pi)$ and let $a$ be a real number. Assume $X$ and $Y$ are independent . Find the density of $Z=X+a \cos(Y)$. I ...
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### Let $X$ and $Y$ be square integrable random variables s.t. $E[X|Y]=Y$ and $E[Y|X]=X$. Prove $P(X=Y)=1$.

Let $X$ and $Y$ be square integrable random variables s.t. $E[X|Y]=Y$ and $E[Y|X]=X$. Prove $P(X=Y)=1$. Furthermore, when the condition changes to $X$ and $Y$ are integrable, show that the conclusion ...
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### In how many ways can weak law of large number can be proved? [on hold]

Can anyone tell me in how many ways can weak law of large numbers can be proved?(Except using Chebyshev's theorem )
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### Proving $E[X]=\sum_{i=1}^{n}P(X\geq i)$ for a R.V which receives non-negative values

I'm refreshing my knowledge in probability and I cam across the following: Let $X$ be a discrete R.V that takes only non-negative values, then $E[X]=\sum_{i=1}^{n}P(X\geq i)$ I have a small ...
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### Distribution determined by its cgf

It is well known, that if the domain of the mgf $M:=E[e^{uX}]$ of a random variable $X$ contains an interval around zero, then the distribution is completely determined by its moments. Consider the ...
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### Sampling conditional random variables

I am going to generate a sample of random variables conditioning on a linear constraint. To make it clear, suppose that I want to generate multivariate gaussian $(0,\Sigma)$ conditioning on the plane ...
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### 3 card monte carlo variation

A friend wants to play a betting game with you. There are 3 upside-down cards on the table 2 black and 1 red. Your job is to find the red card. For every dollar you bet he will give you 2 to 1 odds (i....
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### Compare two coin tossing games

Compare the following two games: You have a fair coin. After one toss, you will get 1 dollar if you get a head, and 0 dollars if you get a tail. How much will you be willing to pay to play this game ...
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### Intuition Behind this Theorem About Brownian Motion

I am having a hard time with the intuition behind some of the representation theorems dealing with Brownian Motion. I think if someone can simply explain the intuition behind this theorem then ...
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### Infinitesimal generator of Brownian motion with additional jumps

A compound Poisson process is a jump process with two parameters, the rate of the jumps $\lambda$ and the distribution of the jumps $\mu$ ($\mu$ is a probability measure on $\mathbb{R}$). The ...
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### Coin Flipping - Probability and Value Proposition

Rusty with probability here... The question is: Flip a coin 11 times. If you get 8 tails or less, I will pay you \$1. Otherwise, you pay me \$7. Step 1. Find the expected value of the ...
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### Conditional mutual information for continuous random variables

Cover and Thomas provides definition of Conditional Mutual Information (CMI) for discrete random variables but doesn't say anything about continuous variables. Wikipedia has a section about a "more ...
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### Probability that the second throw of a fair die exceeds the first

A player throws an ordinary die and records the score $A$. The player then throws the die again and again records the score, $B$. if $B>A$ then we set a score for this player. What is the ...
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### Supremum of expected value over equivalent measures

I'm not sure how one can proof the following statement: We have a probability space $(\Omega, \mathbb{F}, \mathbb{P})$ and a $\mathbb{F}$-measurable random variable $X$. Furthermore we have a set of ...
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### Example of Markov property

I am reading Durrett's Probability: Theory and Examples, and trying to understand its context about Markov Chains. It is not hard to understand the theorem and proofs but when it comes to concrete ...
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### How can I prove that for a random variable $X$, we have $P(X \le \mu) = P(X \ge \mu)$? [on hold]

Specifically, suppose that $X$ is a random variable with properties $\mathrm{Var}(X) = 9$ and $\mu = \mathbb{E}(X) = 2$. How can I prove the following? $$P(X \ge \mu) = P(X \le \mu)$$ It is also ...
Let $Z_1, Z_2, Z_3$ be independent standard Normal R.V.'s. Which of the following has a Chi-Square distribution with 1 degree of freedom. \begin{align} A) & & & \frac{Z_1^2, Z_2^2}{2} ... 1answer 33 views ### Is it true that E(X_1\mid X_1+X_2=k+1)−E(X_1\mid X_1+X_2=k)≤1? I was wondering if we can show that E(X_1\mid X_1+X_2=k+1)−E(X_1\mid X_1+X_2=k)≤1 in general? Here X_1 and X_2 are independent but may not follow the same distribution. Any hint is much ... 1answer 17 views ### When is the sum of uncorrelated (not necessarily with the same distribution) r.v.'s bounded in Probabilty? Let v_{i},\;i=1,\cdots,N be such that E\left(v_{i}\right)=0, E(v_{i}^{2})=1 and E\left(v_{i}v_{j}\right)=0\;for\;i\neq j. So v_{i}'s are mean zero with unitary variance, uncorrelated and ... 0answers 26 views ### distribution and density of maximum minus element I am a bit rusty in probability, and for a project I am studying the random variable Z = \max(X_1, \ldots, X_n) - X_i, i = 1, \ldots, n where the X_i are positive independent random variables. In ... 1answer 26 views ### E(X_1|X_1+X_2=k) increases with k? X_1 and X_2 are independent, but they may not follow the same distribution. I want to know whether E(X_1|X_1+X_2=k) increases with k. I guess this is correct, but is there a proof or counter ... 0answers 11 views ### Gram matrix for a random variable vector space with inner product? I am wondering if it is possible to construct a list of binary valued random variables, \{\bf{X}_1,\bf{X}_2,\bf{X}_3\} and define a Gram-like matrix like \begin{bmatrix} \langle\bf{X}_1,\bf{X}_1\... 0answers 14 views ### Prove \frac{\partial}{\partial\rho}P(X_1 > 0, X_2 > 0) =\frac{1}{2\pi\sqrt{1-\rho^2}} [closed] Let X_1,X_2 have a bivariate normal distribution with zero means, unit variances, and correlation \rho. Show that \frac{\partial}{\partial\rho}P(X_1 > 0, X_2 > 0) =\frac{1}{2\pi\sqrt{1-\...
Let, $(\Omega,\mathcal{F},P)$ be a probability space and $X:\Omega \rightarrow \mathbb{R}$ be a random variable. Let $F_{X} (x)$ be the cumulative distribution function of $X$. Show that if $F_{X} (x)$...