If $X$,$Y$ are independent random variables, $\mathbb{E}|X + Y|$ is finite...
As the title suggests, I am trying to prove$$ E|X + Y| < \infty \implies E|X| < \infty $$for independent $X$ and $Y$ (but not necessarily equal in distribution). Triangular inequality goes the...
View ArticleConditional expectation on function
Under which condition on $f$ we have $E(Y|X,f(Z, W)) = E(Y|X, Z, W)$ for random variables Y, X, Z, W? This seems to hold when f is injective, since then $f(Z, W)$ and $Z, W$ give the same information....
View ArticleFollow-up to Expected-Value Dice question
Problem: Given that the first 6 occurs before the first 5, what is the expected number of rolls of a fair six-sided dice it will take to roll a 6 for the first time and stop?This question was asked...
View ArticleCalculating average number of tries for an event whose probability increases...
I have a question that I didn't even know how to word on google.I have a system where an event has a 2% chance of happening each roll for the first 50 rolls, and, starting from the 51st, its...
View ArticleFinding Optimal Conditional Distributions for Varying Values of
Consider the following expression:$$\sum_{(x, \bar{x})} Q_X(x) Q_X(\bar{x}) \log \sum_y P(y |x)\left(\frac{T(\bar{x} | y)}{Q_X(\bar{x})}\right)^{s},$$where $Q$ is a distribution over $X$, and $P$ is a...
View ArticleProof of Stein's Lemma. Why is this last term zero? [duplicate]
I must be overlooking something simple. I found these notes online Lemma 3.6.5 and I cannot understand why the last term in Here is 0...I will restate the Lemma from the paper below with slight...
View ArticleWhat is the real "random"?
Let's think about a question:Suppose $a_1 ,a_2, \cdots,a_n\in\mathbb{R^+}$, such that $a_1 +a_2+ \cdots+a_n=1$. Then randomly select a set of non-negative real numbers:$b_1 ,b_2, \cdots,b_n$, such that...
View ArticleAssignment Problem, exponential random cost matrix
A group of 𝑛 people are to be assigned to a set of 𝑛 jobs, with one person assigned to each job.For a given set of 𝑛2 values 𝐶𝑖𝑗, 𝑖, 𝑗 = 1, ⋯ , 𝑛, a cost 𝐶𝑖𝑗 is incurred when person 𝑖 is assignedto job...
View ArticleExpectation of odd functions for mean-zero Gaussian processes
Is it true that for any Gaussian process $x(t)$ governed by a stochastic differential equation, such as the Ornstein-Uhlenbeck (O-U) process, with mean zero, the expected value of any odd function...
View ArticleExpected Number of Books to Examine Before Drawing All 6
Tom is searching for the 6 books he needs in a random pile of 30 books. What is the expected number of books must he examine before finding all 6 books he needs (PUMAC Combo Round 2007)?I'm just...
View ArticleProspect and expected utility theory
I have two questions on my homework that I got wrong and don't understand what the right answer should be:Scenario:Suppose you've just spent 800 dollars on a new laptop and have the option of getting...
View ArticleVariant of Peas Floating on Top of Glass question
The original question is here: Probability for six peas to be floating at the top of the glass?I was interested in the following variant of the question, and whether or not my solution was...
View ArticleWhat is the expectation of this power series of random variables?
Let $(a_k)_{k\in\mathbb{N}}$ be iid random variables distributed uniformly in $(-1, 1)$, and consider, for some fixed $r \in [0, 1)$, the limiting random variable$$ X = \lim_{n \to \infty} \sum_{k=0}^n...
View ArticleGiven $n$ natural numbers, if each of them is perturbed, what is the expected...
Suppose you have $n$ natural numbers $a_1 = 0$ and $a_2,...,a_n \in \{0,1,...,2^n\}$, and the corresponding $n$ random variables $r_i \sim N(a_i,\sigma^2)$ for each $i \in [n]$ (where $N(x,\sigma^2)$...
View ArticleWhat is the expected number of unique samples when sampling with replacement...
Given a set $S$, where each member $s_i$ has a probability to be sampled $p_i$. What is the expected number of unique samples when randomly sampling $n$ items with replacement?My question is similar to...
View ArticleExpected value nightmare
Ok, nightmare may be an exaggeration, but for someone not versed in stochatics, it gets close. I have the following two "composite" random variables:$$\frac{r_{k}}{r_{k}+g_{k}} \text{ and }...
View ArticleDoes this probability theory problem have a name?
I found the following problem in a step of a proof of a certain theorem.Given a random variable $X$ with bounded expectation and $\alpha\in(0,1)$, the problem is$$\min_{z\in...
View ArticleExpectation of vectors and matrices' expressions with selector vectors
I need to simplify the expression given here so that it comes in terms of mean, variances and size of the vectors and matrices involved in it.$\operatorname{Tr} \left( \mathbb{E} \left\{ e_n^H \cdot...
View ArticleExpectation of a bivariate function vs the expectation of the transformed...
Let $(X_1,X_2)\sim f(x_1,x_2)$, also $\phi=g(x_1,x_2)$, where $g$ is invertible with respect to both $x_1$ and $x_2$, we can obtain the density of $\phi\sim p(\phi)$ through the Jacobian rule.My...
View ArticleExpected number of consecutive sequences
Let's say we have a string of length n and each character has probability p of being 1 and probability 1-p of being 0. What is the expected value of the number of "islands" of 1s in the string (the...
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