If $E(Y)=0$, $Cov(Y, E(Y|X))\geq0$ for any random variable $X$
Let $Y$ be a random variable with mean zero and finite variance, and let $X$ be any other random variable. Using the fact that $\mathbb{E}[Y]=0$ and the law of iterated expectation, we have...
View ArticleDefinition of conditional expectation with two random variables.
Most definitions of the conditional expectation use $\sigma$-algebra arguments:Let $X$ be a random variable on a probability space $(E, \mathcal{E}, \mathbb{P})$ with $\mathbb{E}| X|\le\infty$and...
View ArticleConvergence of Conditional Expectation of Convergent Random Variables with...
I have been attempting to solve the following question with no luck for a while now: Let $X_1,X_2,\dots \in L_1$ with $X_n \uparrow X$ a.s. (and $X \in L_1$) . Show that for any filtration $(F_n)$ we...
View ArticleCan an IID sequence be a martingale?
I was looking into Doob's upcrossing inequality, which says that for a supermartingale $X$ and real numbers $b > a$, $$(b-a)\mathbb{E}[N_n([a,b], X)] \leq \mathbb{E}[(X_n -a)^-],$$where $N_n([a,b],...
View ArticleFind the conditional expectation $E[X \mid X \leq p]$
Let $X$ be a discrete random variable that can take values from 1 to $n$, where $n$ is a large fixed number and also let $p\ll n$ be a fixed number. I am trying to find the expectation $$E[X \mid X...
View ArticleUnder what assumptions $Y_n \xrightarrow{L^1} 0$ as $n \to \infty\ $?
Let $\{X_n, U_n\ |\ n \geq 1 \}$ be a collection of independent random variables such that $X_n$'s are iid random variables following standard Cauchy distribution and $U_n$'s are independent random...
View ArticleFormula related with the partition theorem
I am stuck with solving this problem:Let $X$ be a continuous random variable with probability density function $f_X$. Then,$$ f_X(x)=f_{X\mid X<0}(x)\cdot P(X<0)+f_{X\mid X\ge 0}(x)\cdot P(X\ge...
View ArticleHelp with the probability generating function for a conditional distribution.
I am trying to follow the following proof from Gut's book An Intermediate Course in Probability.Let $X$ and $N$ be random variables.$N \sim Po(\lambda)$ and $X|N=n \sim\operatorname{Bin}(n,p)$ the...
View ArticleExpected value of a Markov Chain, multiplied at different times.
Let $(x^n)_{n\geq 0}$ be a Markov chain indexed by $n$, let $\mathbb E_{\mu}$ denote the expectation taken such that the initial condition is distributed as $\mu$. I can see heuristically why is should...
View ArticleVariance of product of two random variables where one is a Bernoulli whose...
I have the following problem. I have a Gaussian variable $x$ with mean $\mu$ and variance $\sigma$ and I construct a certain function $0 < f(x) < 1$. This function value becomes then the...
View ArticleConditional expectation and independence with respect to $\sigma$-algebra and...
This question concerns an extension of a previously solved question regarding measure-theoretic definitions of conditional expectation and conditional independence with respect to a $\sigma$-algebra...
View ArticleLaw of total expectation and recursion
I have a problem with one exercise from Probability and Random Processes (2001) by Geoffrey R. Grimmett and David R. Stirzaker.A coin shows heads with probability $p$. Let $X_n$ be the number of flips...
View Articleconditional expectation on uniform RV [closed]
Consider $X,Y,Z$ independently and identically distributed from uniform distribution $[0,1]$.I am trying to solve for:$E(X|X>Y)$.$E(X|X>Y,X>Z)$.I am just tyring to understand these simplace...
View ArticleIs $\mathbb{E}[X|\mathscr{B}]$ well-defined when $\mathbb{E}[X|\mathscr{A}]$...
Let $X$ be a random variable on a probability space $(\Omega,\mathscr{F},\mathbb{P})$,and let $\mathscr{A}$ and $\mathscr{B}$ be sub-$\sigma$-fields of $\mathscr{F}$ such...
View ArticleConditional expectation of length of path
We have $n$ houses, $n\geq 2$, that are set in a straight line, such that the distance between any two neighboring houses is $a\in (0, \infty)$. A resident from a randomly chosen house goes on a visit...
View ArticleConditional expectation of a Brownian motion wrt the minimum of its past...
Does someone have an idea to deal with the below expectation$E(W_t| S_t>-1)$with $S_{t} =min_{s<t}W_s$With W a standard brownian motion.I tried to use the reflection principle or make appear some...
View ArticleLe Gall Exercise 11.3 conditional expectation and stopping time
I am trying to solve the exercises 11.3 from the book Le Gall: Measure Theory, Probability and stochastic processes.Exercise 11.3: Let $(Xn)$ be a sequence of independent real random variables...
View ArticleCalculate the Variance of $\min(N_k,p)$
I am trying to compute the variance of a random variable$\min(N_k,p)$, where $N_k$ is a random variable and $p$ is a fixed number. I have computed the expectation using the conditional expectation,...
View ArticleExpected value of $X$ given $a^TX$ if X has a multivariate normal distribution.
Let $X$ be a normally distributed random vector with mean vector $\mu$ and covariance matrix $\Sigma$. Also, let Y be a random vector such that $Y = a^TX$ where $a$ is a constant vector. How would I go...
View ArticleWhy is $T = E[T_1 + ... + T_n | T]$ where $T = T_1 + ... + T_n$ sum of i.i.d....
Im trying to understand the solution of an exercise.Let $T_1, ..., T_n$ be iid Poisson random variables with parameter $1$ and set $T:= T_1 + ... + T_n$. What is the law of $\mathbb{E}[T_1 | T]?$In the...
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