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Questions tagged [expected-value]

Questions about the expected value of a random variable.

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If $X$ ~ $\text{Beta}(\alpha, \beta)$, find $\Bbb{E}\left[ X(1-X)\right]$

I am currently reading through John I.Marden's Mathematical Statistics: Old School on my own and I am trying to solve the following problem about expectation. If $X$ ~ $\text{Beta}(\alpha, \beta)$, ...
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Expected value of a lognormal random variable formula

Given $Z(t) \sim \mathcal{N}(0,\,1)$, why does the formula for the expected value of a lognormal random variable, $E(e^{X}) = e^{\mu + \sigma^2/2}$ give the following: $$E(e^{\sigma \sqrt{t} Z}) = e^{...
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Problem with proving simple martingale property

I want to prove the following property : Let $(X_n)$ and $(Y_n)$ be a martingale with respect to the filtration $(F_n)$. Let's also assume that $E(X_n^2),E(Y_n^2)<+\infty$ Prove that $$E(X_{n-...
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Finding expected value when conditional distribution is known

If the distribution of $Y$ conditional on $X=x$ is known, and the distribution of $X$ is known, what would be the general process for finding the expected value $\Bbb E[Y]$? IS there a general ...
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$Y = \frac{X_1 X_2}{X_3}$ where $X_i$ is a uniform random variable

$Y = \frac{X_1 X_2}{X_3}$ where $X_i\sim U(0,1)$ and $X_1,X_2,X_3$ are i.i.d I need to calculate $Var(Y)$ and $Var[Y|X_3=1.7]$ I know that for each $X_i$, $E[X_i]=\frac{1}{2}$ $Var[X_i]=\frac{1}{...
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52 views

Tossing Coin Expected Value Against Intuition

I have the following two coin toss games: Game 1: A and B tosses a coin. At first the coin is unbiased. Through the game, if heads comes A wins and game stops. If tail comes, the coin is swapped with ...
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Proving expectation and variance of a function of a random variable tends to a fix point

Given $f:\mathcal{X} \rightarrow \mathbb{R}$ is a continuous function and $\mathbb{E}_{Q(X)}[X] \rightarrow x^\star$ ($x^\star$ is a fix number), $\mathbb{V}\text{ar}_{Q(X)}[X] \rightarrow 0$. How can ...
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Bound for the expectation of a function of a Gaussian random vector

A paper I am currently reading seem to be using the bound $$ E_h\left[\inf_{\|z\|_{\infty}\leq t} \|z-h\|^2\right] \leq n\frac{1}{\pi}\frac1t e^{-t^2/2},$$ where $n$ is the dimension of the vector, ...
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Expectation of the product of Brownian processes (higher powers)

I have recently sat an exam that had elements of stochastic calculus, but I am now feeling like I might have gone wrong in some questions of it like the following. I am trying to evaluate $\mathbb{E}(...
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Proving convergence of expectation and variance given Rényi's $\alpha$-divergence tends to 0

I denote $p, q$ as density function of $P, Q$. Given $Y, X$ are random variables and \begin{align} \int q(x)\mathbb{D}_{\alpha}[p(Y\mid X=x)\,||\,p(Y\mid x^\star)] \,dx \rightarrow 0 \end{align} ...
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Can Conditional Expected Value be negative in normal distribution?

So, the problem gives me this facts (for a Normal bivariate distribution X,Y) $$Var(Y|X=x) = 5$$ $$E(Y|X=x) = 2 + x$$ It asks me to find $$E[Y^2|X=7]$$ I tried this: using the conditional variance ...
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Expected Value with 2 coin flips performed multiple times

I am trying to make sense of Expected Value. Assume coin flips. Each trial is two coin flips: { HH, TT, HT, ... } The probabilities are: P(H) = 1/2, P(T) = 1/2 The trial is repeated N = 8 times ...
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Weibull distribution -variance, expected value [on hold]

If $X \sim Exp(\lambda)$, so what is a variable distribution of $Y=X^{\frac{1}{\alpha}}$ for $\alpha >0$? Calculate the expected value, variance and intensity of failures. For which the value of $\...
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+100

Best way to visualize queueing theory for a Lecture on Markov Chains

Let the Markov Chain $X:=(X_{n})_{n \in \mathbb N_{0}}$ denote, for every $X_{n}$, the number of people waiting in a line at time $n$. Now note that $X$ is a Markov Chain living on $\mathbb Z_{+}$. ...
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conditional probability about gambler winning x amount of coins

A gambler plays seven games one after the other and the chance to win each of them is $\frac{1}{3}$, independently of the others. For $k = 1, ..., 7$, if the gambler wins game number $k$, then the ...
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“expectation of sum is sum of expectation”, is this claim true? if yes, how to justify this claim?

this post is saying linearity of expectation gives following equation $$\mathbb{E} [\sum_{j\neq i} Y_i Y_j] = \sum_{j\neq i} \mathbb{E} [Y_i Y_j]$$ per wiki, Linearity of Expected_value is ...
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Why does the summation of these indicator variables start from i<j?

I'm currently reading through the eighth edition of A First Course in PROBABILITY, by Sheldon Ross. The section I'm reading is "Momens of the number of events that occur", and I understand everything ...
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Expected Value: What exactly did I do?

Schwin has a large jar containing some M&Ms, each with the letter "m" stamped on it. He removes 1000 candies from the jar, and removes the letter "m" from each one. He then returns all of the M&...
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Finding the probability of success that maximizes the variance of independent trials

A professor wishes to make up a true-false exam with n questions. She assumes that she can design the problems in such a way that a student will answer the jth problem correctly with probability $...
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Actual meaning of the formula $E[\phi(X,Y)|\mathcal{G}]= E[\phi(x, \cdot)]|_{x=X}$

Let's suppose we have a probability space $(\Omega,\mathcal{F},P)$ and a sub sigma algebra $\mathcal{G} \subset \mathcal{F}$. Let's say we have $X$ that is $\mathcal{G}$-measurable and $Y$ that is $\...
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Find mean and variance from mgf where t is denominator

For continuous random variable X, pdf: $f_{X}(x)=2(1-x), x\in[0,1]$ mgf: $M_{X}(t)=\frac{2(e^t-t-1)}{t^2}$ Problem is to find mean and variance from mgf, I tried using $\frac{d}{dt}M_{X}(0)$ and $\...
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how to derive this property of exponential distribution?

Let $X$ be a real random variable with exponential distribution on a measure space with probability measure $\Bbb P$. Let $\Bbb E$ be the expected value. Then \begin{equation}\Bbb P(X>s+t|X>s)=\...
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When is a linear recurrent process stationary?

Let’s call a sequence of random variables $\{X_n\}_{n = 1}^\infty$ stationary, if $\forall n, m, k \in \mathbb{N}$ $EX_n = EX_m$ and $Cov(X_n, X_m) = Cov(X_{n + k}, X_{m + k})$. Let’s call a sequence ...
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Expected value in a linear combination

I have a random variable Y, that is defined by: $$Y = aX_1 + bX_2$$ Where we know $X_1 $ and $X_2$ are independent. How do I write out $EX_1$ and $EX_2$ in terms of only a, b, EY, and VarY? I ...
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Showing $E(X) = \sum_{i}E(X\mid A_i)P(A_i)$

Following is my proof. Suppose $X$ is a discrete-type random variable ranging in the set $S$ and $\{A_i : i=1,2,3,\dots\}$ is a finite or countably infinite partition of a sample space $\Omega$. We ...
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Expectation of number of hubs in a random graph

Suppose $\Gamma(V, E)$ is a finite simple graph. Let’s call a vertex $v \in V$ a hub if $deg(v)^2 > \Sigma_{w \in O(v)} deg(w)$. Here $deg$ stands for the vertex degree, and $O(v)$ for the set of ...
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Finding the expected number of a certain colored ball drawn from an urn in k draws

Suppose we have an urn containing c yellow balls and d green balls. We draw k balls, without replacement, from the urn. Find the expected number of yellow balls drawn. Hint: Write the number of ...
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Calculating $E(2W(s)+W(u)|W(u)=2)$

Let $W(t)$ be standard Brownian motion and let $u<s$. I know that $W(s)\sim\mathcal{N}(0,\sqrt{s}), W(u)\sim\mathcal{N}(0,\sqrt{u})$ and $2W(s)+W(u)\sim\mathcal{N}(0,\sqrt{8s+u})$. How should I ...
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$X$, $Y$ i.i.d r.v's. Prove that $\mathbb{E}[X\mathbb{1}_{\{X+Y \in B\}}] = \mathbb{E}[Y\mathbb{1}_{\{X+Y \in B\}}] $

Let $X, Y$ be i.i.d random variables with finite expected values. I want to justify that $$ \int_{\{x+y \in B\}}x\mu(dx)\mu(du)=\int_{\{x+y \in B\}}y\mu(dx)\mu(du). $$ I would appreciate any hints, ...
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Expectation of dependent Bernoulli sum

I want to estimate the expected value of the following sum of random variables, $$ Y = \sum_{i=1}^N X_i $$ where each $X_i$ is a Bernoulli random variable. In particular, $$ X_1 = \begin{cases} 1, &...
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What's the equation for $\mathbb{E}(XY)$?

Let $X$ and $Y$ be independent, discrete random variables. Suppose I want to find $\mathbb{E}(XY)$. What's the equation for it? My educated guess is that the equation for the expected value in this ...
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Expected Value for weighted choices without replacement

I have a game where I will choose some number of prizes from a weighted pool of $n$ prizes. If I choose prizes from this pool while replacing the prizes I had already chosen, I would have an Expected ...
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Expected value of multiplication of matrices

While reading through Xu et al. (2016) I stumbled upon this proof: $$ \begin{align} \mathbb{E}[Wyy^\intercal W^\intercal] & = \mathbb{E}[Wyy^\intercal W^\intercal-W\mathbb{E}[y]\mathbb{E}[y]^\...
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How do I make sense of this expectation?

I am having some difficult in making sense of ${\rm{E}}\left[ {\int\limits_0^t {{W_u}du} } \right]$ where W is just your standard one dimensional brownian motion. Is interchanging the order of the ...
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100 tickets given to guests each having p probability of winning? what is the expected number of guests who win the prize?

A host gives a ticket to each of the $100$ guests at his party. Each ticket has the probability $p$ of winning some prize (independent of the others). (a) What is the probability that exactly $10$ ...
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Bernstein-Chernoff inequality

I am studying the proof of the following result due to Bernstein and Chernoff: Let $\xi_1,\ldots,\xi_m$ be independent random variables with $|\xi_i|\leq 1$ for $i=1,\ldots,m$. Let $\beta = \sum_{i=1}...
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Accuracy of Euler Monte Carlo discretization without knowing exact solution?

By using Euler Monte Carlo discretization (for a Hull-White model) we simulate $$r(t+\Delta t)=r(t)+\lambda(\theta(t)-r(t))\Delta t+\eta\sqrt{\Delta t}Z$$ with $Z\sim N(0,1)$, $\lambda$, $\eta$ ...
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Expected value of conditional probability event

This is in continuation to Expected value of conditional events In a cricket match, event A is a subset of all possible moves taken up by the fielding team. The batsman has a set of moves, and, ...
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Trouble with seemingly extremely simple statistics question involving normal distribution and expectation.

Say we have a random variable X which is distributed like such: X ~ N(1, 4). A question asks me to calculate $E(X^2)$, which I thought would be straight forward. I use the formula for Variance: $Var(...
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Cauchy-Schwarz inversion like inequality for expectactions of comonotonic functions

Given two non constant, integrable, comonotonic functions $x_1, x_2\colon [0,\infty) \to [0,1]$, i.e., both functions are non decreasing or non increasing, I need to prove that $$\big(E[x_1(T)]+E[x_2(...
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A coin is tossed until a head turns up and $\$f(n)$ is paid out, find the expected value of the payment

A fair coin is tossed until heads turns up for the first time. If heads turns up for the first time on the nth toss you receive $2^n$ dollars. (a) Show that the expected value of your winnings ...
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Finding the expected winnings of the martingdale doubling system

In the martingale doubling system the player doubles his bet each time he loses. Suppose that you are playing roulette in a fair casino where there are no 0's, and you bet on red each time. You then ...
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Calculating variance of a sum

The number of students per day has the distribution N ∼ Poisson(10). The students of CSUEB withdraw money from a cash machine according to the following probability function (X): X | 50 | 100 ...
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Helping to understand the expected value and variance of an estimator $\hat{X}\left(y\right)=\frac{2}{N}\left(y_1,y_2,…,y_n\right)$

I have found with the following example: Suppose that we have samples y drawn from a uniformly distributed random variable which extends from 0 to x. We would like to estimate x from the data $y_1,......
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Expected Value of Exponential Function of a Single Variable

Given that I know: 1) $E(n)=2$ (or expected value of $n$ is $2$) 2) $n$ must be a positive integer ($n=1,2,3,\ldots$) I am just wondering is there any method to evaluate $E(2^{n-1})$ (the expected ...
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Prove that $X_n \to X$ in $L^{2r}$ and $Y_n \to Y$ in $L^{2r}$ then $X_nY_n \to XY$ in $L^r$

My question concerns the following statement: Suppose that $X, Y \in L^{2r}$ and $X_n \to X$ and $Y_n \to Y$ in $L^{2r}$. Show that $X_nY_n \to XY$ in $L^{r}$. hint: $(a+b)^r \leqslant 2^{r-1}...
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Distribution of expectation operator when computing mgf of X bar

I'm trying to work through the proof for the moment generating function of $\overline{X}$. The proof below looks fairly straightforward but I'm having trouble understanding getting from the 2nd to ...
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Having trouble determining the sample space for conditional expectation problems

My book states the following theorem Let $X$ be a random variable with sample space $\Omega$. If $F_1, F_2, . . . , F_r$ are events such that $F_i$ and $F_j $ are disjoint (for $i$ not equal to $...
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Easiest way to see this expectation?

Say you have some normal random variable $X \sim N\left( {0 ,{\sigma ^2}} \right)$ and need the expectation ${\rm{E}}\left[ {{\rm{exp}}\left( {\gamma {{\rm{X}}^2}} \right)} \right]$. Any quick ways to ...
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Example of function $g(x)$ s.t. $E(g|X_n|) \not\to 0$

Consider the following result (assuming probability space ($\Omega, \mathcal F, \mathsf P$)) If $g : [0, \infty) \to [0,\infty)$ bounded, strictly increasing, $g(x) > 0$ for $x > 0$ and $\...