Defining the expectation of a measurable function with respect to a...
The typical definition of expectation requires a probability space and a random variableLet $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space, $(\mathsf{X}, \mathcal{X})$ be a measurable space...
View ArticleExpectation of inner product
Given random vectors $X,Y\in\mathbb{R}^n$ such that $E(X)=0$. Does it means that $E(\left\langle X,Y\right\rangle)=0?$ I am asking this because when I am reading the proofs for the convergence of the...
View ArticlePower of multiplication of two Gaussian distributed matrices
There are two Gaussian matrices, $A\sim N(0,\sigma_A^2\mathbf{I})$ and $B\sim N(0,\sigma_B^2\mathbf{I})$, respectively.Their dimensions are $A\in \mathbb{R}^{L\times M}$ and $B\in \mathbb{R}^{N\times...
View ArticleExpectation of maximum of n i.i.d random variables
I have $n$ i.i.d. random variables, $X_1,..., X_n$ which follow some arbitrary distribution. Based on experiments in Python with various distributions, it seems that $\mathbb{E}(\max(X_1,...,X_n))$ is...
View ArticleSearching a convergent analythical solution to...
How I discover If this expression is convergent:\begin{equation}\mathbb{E}\left[\operatorname{sech}^{2}\left(\alpha \exp x\right) \exp x \operatorname{sign}(x)...
View ArticleComputing conditional expectation of recursive process
Assume that $u(c_s)$ is some concave function. The goal is to evaluate the following recursive stochastic process $V_t$ given by$$V_t = E_t[\int_t^{\infty} (u(c_s) - \beta V_s)ds] $$ for some constant...
View ArticleWrite $\mathbb{P}^X(A)$ as $\mathbb{E}[1_A X]$
$(\Omega, \mathcal{F}, \mathbb{P})$ probability space$(E, \mathcal{B}(E))$ a measurable space where $B(E)$ is the Borel sigma algebra and $E$ is a Banach space$X:\Omega\to E$ random variable with...
View ArticleExpectation of a function of the CDF of a Normal variable
Let X $\sim$$\mathcal{N(\mu, \sigma^2})$. Find the Expectation of $\left(-log_{e} \left(\Phi\left(\frac{X - \mu}{\sigma}\right)\right)\right)^3$ , where $\Phi \left(. \right)$ denotes the cumulative...
View ArticleFinding $MSE$ of optimal estimator
Background of the question:We know that $X$ is a continuous random variable that has $P\left(-1\le X\le 1\right)=1$and $f_X\left(x\right)<\infty $We define $X(n)=cos(\pi n X)$$E[X]=\mu$ and...
View ArticleOptimal Strategy / Stopping Time for repeatedly rolling a die with...
I am taking a college probability theory class, but I have never done anything with optimal strategy / stopping time before and am curious about how I can learn. This is a problem I would like to know...
View ArticleHow to calculate the expectation of this Poisson-like process?
Question: Let $\tau_i\sim \text{Exp}(\lambda_0)$ iid and $\gamma_i\sim \text{Exp}(\lambda_1)$ iid and independent of each other and set $N_t=max\{k\geq 0: \sum_{i=1}^k{(\tau_i+\gamma_i)}\leq t\}$. What...
View ArticleHow to find $E(X)$ and $E(X^2)$ for a normal distribution from known values...
I am self-studying an introductory course on probability and am attempting some questions to test my understanding. As I have only just started I am confused on where to begin with a problem like...
View Articleapproximating expectation of non-linear transformation of stochastic process...
I have a simple DISCRETE TIME $t\in[1,2,3,4,5,6,7,8,9]$ stochastic process$$f(t)=f(t-1)\cdot\mathrm{e}^{\mathbf{X}_t}$$where X is a i.i.d. random normal variable$$\mathbf{X}_t \sim...
View ArticleTrying to find the conditional expected value of X when X + Y = m, and X, Y...
If X and Y are independent binomial random variables with identical parameters n and p, calculate the conditional expected value of X given X+Y = m.The conditional pmf turned out to be a hypergeometric...
View ArticleMonotonicity with expectation
I think the following is true but I cannot prove it.Let $Z_1, Z_2$ are two random variables defined on the same sample space $\Omega$. Suppose that $Z_1(\omega) < Z_2(\omega)$ for all $\omega\in...
View ArticlePower average of multinomial distribution
Given a multinomial distribution $X=(X_1,\dots,X_K)$, $\sum_{k=1}^KX_k=n$, and $p_1=\dots=p_K=\frac{1}{K}$, I wonder whether there is some approximation or lower bound (with respect to $K$, $n$, and...
View ArticleCalculating Expected Throws for All Die Faces to Appear Twice [duplicate]
I'm working on a problem where I need to find the expected number of throws required for each face of a six-sided die to appear at least twice. I've conceptualized the problem using a triplet $(a, b,...
View ArticleOn the relation of the expected squared norm of a random vector with its...
Let $\mathbf{x}=(x_1,\ldots,x_n)^\top\in\Bbb{R}^n$ be a random vector with mean $\mathbf{\bar{x}}=(\bar{x}_1,\ldots,\bar{x}_n)^\top\in\Bbb{R}^n$ and covariance matrix $\Sigma\in\Bbb{S}_{++}^n$, where...
View Article$E[\exp(-sX)\exp(-sY)]$ for two identically distributed, but correlated...
I am trying to figure out the following problem. I am trying to evaluate the expectation: $E[\exp(-sX)\exp(-sY)]$, where $X$ and $Y$ are identically distributed, but correlated random variables, hence,...
View ArticleBayesian updating on expectation with many candidates
I have an individual selected for a job. He was competing against 10 candidates. All individuals are independently drawn from uniform distribution U(0,1). Him being chosen means he is better than the...
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