# 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 $X$ 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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### 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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### 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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### 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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### 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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### 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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### 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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### Covariance Matrix of Uniform Distribution Positive Definite

Suppose that $B$ is a Lebesgue measureable subset of $\mathbb{R}^d$. Let $U$ be the uniform distribution on $B$. Let $x \sim U$, $\mathbb{E}[x] = 0$. and let $M = \mathbb{E}[xx^T]$, be the covariance ...
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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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### Existence of Joint Distribution from Overlapping Marginal Distribution

Suppose $x_i\in \mathbb{R}^{n_i}$ for $i=0,1,...,k$. For each $i=1,...,k$, suppose $F_i$ is a probability measure of $(x_0,x_i)$ on $\mathbb{R}^{n_0 + n_i}$. Assume all $F_i$ have the same marginal ...
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### Proving that a complex expression of integrals is increasing in a given parameter

Let $f$ and $F$ denote the respective pdf and cdf of a probability distribution on $\mathbb{R}$. Consider any natural $n\geq3$ and any real $c$ such that $c\geq0$, and $\rho\geq0$. We want to prove ...
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### In search for a sequence of r.v. with particular conditions

I am looking for a sequence of real random variable $(X_n)_{n\geq 0}$ on a probability space $(\Omega,\mathcal F, \mathbb P)$ such that : $\forall n >0,$ $X_{n+1}-X_{n} <+\infty$ a.s. It ...
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### Calculating the expectation of binomial distribution without calculating the summation

We know that expectation of a binomial distribution is $$\sum _{1}^{n}\left(\begin{array}{c}n\\ k\end{array}\right){p}^{k}{\left(1-p\right)}^{n-k}k = np$$ But while proving it, it is being written ...
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### measure of a set which is a subset of infinitely many subsets of probability measure space [on hold]

Let $B,A_1,A_2,....$ be the subsets of a probability measure space. If $B \subset \bigcup A_j$, show that $m(B) \le \sum_{j=0}^\infty m(A_j)$. I have no idea as how to approach it. I do have the ...
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### Continuous version of a Poisson R.V.

I am wondering if there is a continuous version of a Poisson random variable, that has the following two features: 1) Has a CDF that agrees with the discrete Poisson distribution on the integers, and ...
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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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I have some question about the Monotone Class Theorem and its application. First I state the Theorem: Let $\mathcal{M}:=\{f_\alpha; \alpha \in J\}$ be a set of bounded functions, such that $f_\... 1answer 46 views ### Bayesian Estimation: calculating an integral I am reading a book on Bayesian filtering and I have a question regarding calculating transition density$p(X_t|X_{t-1})$. My question is how the term$p(X_t|X_{t-1}, V_{t}=v)$is converted to the ... 1answer 29 views ### Probabilistic Method/Model for Traffic Flow Context: Given a network system or a traffic system with some value related to the system. Question: Which probabilistic methods, model, distributions are used frequently to predict a event (for ... 1answer 427 views ### What happens to a random walk when we increase the probabilities of going right? Consider a random walk on the integers where the probability of transitioning from$n$to$n+1$is$p_n$(and of course, the probability of transitioning from$n$to$n-1$is$1-p_n$); we assume all$...
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 ...