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Questions tagged [expected-value]

Questions about the expected value of a random variable.

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Expectation of a battery lifetime: Uniform Distributions

Question: A battery has a lifetime of $24$ hours and it is used for maximum three days. On each day, a person uses the battery for $K$ hours, where $K$ is uniform on $[0,24]$ and independent of the ...
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Probability - Expectation value and Covariance

A group of $n = 10$ men exchange their gloves randomly (altogether there are 20 gloves). Let $X_i$ be the random variable which attains the value 1 if the $i-th$ man got at least one of his gloves ...
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The expectation of a geometric random variable where its parameter is uniform

First thanks for any help editing my text. If a random variable $X$ has a geometric distribution with parameter $P$ where $P$ itself is a random variable and uniformly distributed from $0$ to $1-1/n$,...
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Expectation of sum of independent Poisson distributions

I have three independent Poisson distributions: $X_1 \sim \mathcal{P}(15)$, $X_2 \sim \mathcal{P}(21)$ & $X_3 \sim \mathcal{P}(10)$. I wish to find the Expectation and Variance of $X_1 + X_2 + ...
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Intuition behind Expected Value of the Square of a Random Variable $X$

Let's look at the case of $X \sim Pois(\lambda)$. Since $k \in X(\Omega)$, it is clear that $k^2 \in X(\Omega)^{2}$. Following this logic and from an intuitive view I'd say $\mathbb E[X^{2}]=\sum_{k^...
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Does negative correlation survive monotone transformation?

Let $X$ and $Y$ be two non-negative random variables and be negative correlated, i.e., $$\mathbb{E}[XY] \leq \mathbb{E}[X]\mathbb{E}[Y].$$ Let $h(\cdot)$ and $g(\cdot)$ be two non-negative, monotone ...
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Expectation and RIP question

Hi I am trying to solve the following question but couldnt figure out if i am moving in the right direction or not. The question is as follows. I have a large number of candies. One day i decided to ...
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Question about book solution to estimate $E[e^{XY}]$ when $X$ and $Y$ are independent exponential RVs with $\lambda = 1$

Let $X$ and $Y$ be independent exponential random variables with mean 1. (a) Explain how we could use simulation to estimate $E[e^{XY}]$. (b) Show how to improve the estimation approach in part (a) by ...
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Expected value of the product of dependent Normal random variables

I have 3 independent Normal Random Variables: $A$, $B$ and $C$, each with mean=$0$ and Variance $1$. Then I have $X=3A+5B$ and $Y=A-C$... because both of them are functions of $A$, we know they ...
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Use of law of total expectation without checking integrability

$\newcommand{\E}{\mathbb E}$In basic probability classes, people often use the formula, namely the law of total expectation $$\E[X]=\E[\E[X\mid Y]]$$ without checking integrability of $X$. I can't ...
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Why are there two kinds of formula for Probability Generating Function?

In this link: On the other hand, in this link: So, how are these two equations same?
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Defining Random Variable, Expected Value for Number of Fixed Points given a permutation

Let $n \in \mathbb N$, and $\mathcal{K}$ be the set of permutations possible for a set $\{1,...,n\}$. Let $\sigma \in \mathcal{K}, $ such that $\sigma: [n] \to [n]$ is a randomly selected permutation. ...
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Expected number of slots required for channel access - Birthday paradox

Let $n$ be the number of users who want to access a channel divided into with $s$ time-slots and let $n<<s$. Each user randomly chooses one time-slot (out of $s$) for accessing the channel. A ...
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Expected value when involving intangible factors

Rebecca and James play the following game. James pays her 1.50 for her to roll two dice. If the sum of these two dice is exactly 6, then she will pay him of 10.50 and return him the cost of 1.50 he ...
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buying multiple lotto tickets

Lets say we have a lotto game with 6 numbers to choose from (and you can only select different numbers). The expected gain from one ticket is E(x) = -0.8 and a variance of 1000 . Lets say I want to ...
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proving that $\mathbb{E}[X] = \int_{0}^{\infty}\mathbb{P}(X \geq t) dt$ in the case of non-negative random variables.

say we're in a probability space $(\Omega, \Sigma, \mathbb{P})$ and $X : (\Omega, \Sigma) \to (\mathbb{R}, \mathscr{B}_{\mathbb{R}})$ is random variable I resulted in the right result but I feel like ...
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$\mathbb{E}(X_{Y+1}X_{2}^{2}X_{2}|x_{1})$ with $X\sim N(0,1)$ and $Y\sim Pois(-1)$ both independent

Let $\{X_{i},i\in\mathbb{N}\}$ be a sequence of independent standard normal random variables. Furthermore, $Y$ is a Poisson distributed random variable with parameter $\lambda=-1$, i.e., $\mathbb{P}(Y=...
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Expected Value of a time varying diffusion

I need to compute the following expected Value $$V(w)=\mathbb{E}\left[\left.\int_{0}^{\infty}e^{-rs}W_{t+s}ds\right|W_{t}=w\right]$$ where $W_{t}$ is a GBM whose diffusion is given by $$dW_{t}=\mu ...
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Three questions about the expectation of Brownian Motion. [on hold]

Let W(r) is a standard brownian motion which follows N(0,r). Then, how to calculate (1) $E(\int^1_0W(r)^3dr)$ (2) $E(\int^1_0W(r)dr)^2$ (3) $E(W(1)\int^1_0W(r)dr)^2$ For (1), my solution is $E(\...
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A problem in probability theory about expected value and Fubini's theorem

Let $\{X_{n}:n\geq1\}$ be a seqeunce of square-integrable independent random variables on $(\Omega,\mathcal{F},\mathbb{P})$ with $\mathbb{E}[X_{n}]=0$ for every $n\geq1$. Set $S_{n}:=\sum_{j=1}^{n}X_{...
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Random Incidence Paradox Question

10 friends have a collection of 1000 candies, 15 have collection of 1300 candies, and 5 have collection of 600 candies. What is the average number of candies your friends have? If you mix all these ...
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For a Brownian motion $(B_t)_{t \geq 0}$, do we have $E[(B_\tau - B_\sigma)^2]=E[B_\tau^2 - B_\sigma^2]$ for stopping times $\sigma \leq \tau$?

Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0}, P)$ be a filtered probability space and let $(B_t)_{t \geq 0}$ be a Brownian motion on that space. The question is if the following is true: For ...
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Expectation, variance and conditional probability of combined discrete and continuous random variables

Category: Introductory Probability I have seen many of the other questions with similar titles (there are quite a few!), but unfortunately I am struggling to apply the concepts and knowledge I have ...
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Multidimensional probability expressed as expected value

I have been reading an article and I stumbled upon something that I simply can't work out. Here is a part of the paper: "We denote the probability with respect to $X_n$ by $P_n$, and the expectation ...
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Law of Unconscious statistician - Proof on wiki wrong?

So I don't understand the last part of how wikipedia proves the LOTUS theorem. Here is the link. After we prove that, $F_Y(y) = F_X(g^{-1}(y))$ It says by chain rule we have, $F_Y(y) = f_X(g^{-1}...
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Resolving this summation using the given pmf

So I'm trying to derive an expected value (related to Bayesian risk/loss function) and I've derived all except one final part. To finish the final part I need to derive one of the following expected ...
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Convergence of Expectation of norm of sub-gaussian random vector

1.We know that if $X=(X_1,...,X_n)$ be a random vector with independent sub-gaussian coordinates $X_i$ that satisfy $EX_i^2=1$, then $$||||X||_2-\sqrt{n}||_{\psi_2}\leq CK^2$$ where $K=max||X_i||_{\...
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Expectetion of $Y^{\alpha}$ with $\alpha >0$ [duplicate]

Let $Y$ be a positive random variable. For $\alpha>0$ show that $E(Y^{\alpha})=\alpha \int_{0}^{\infty}t^{\alpha -1}P(Y>t)dt$. My ideas: $E(Y^{\alpha})= \int_{-\infty}^{\infty}t^{\alpha}f_{Y}...
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Manipulation of Functions of Random Variables

Let $X:\Omega\rightarrow\mathbb{R}$ be a random variable such that $\mathbf{E}(X), \mathbf{var}(X)$ exist. a) Consider a function $f:\mathbb{R}\rightarrow\mathbb{R}$ defined: $$ f(t)=\mathbf{E}...
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Expected Length of Arc in Randomly Divided Circle.

Let's say theres a circle of unit circumference m, and the circle is divided by k points all placed randomly. The points will divide the circle up into arcs. What is the expected length of any arc in ...
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Is there an extension of Wald's Equation to the expectation of a product of random variables?

So Wald's Equation states that for a real-values, independent and identically distributed sequence of random variables $(X_{n})_{n\in\mathbb{N}}$ and a nonnegative integer $N$, which is independent of ...
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Coin Tosses and Variance 3 Runs of Heads

A group of $n \geq 10$ people sits at a round table. Each person tosses a fair coin. Let $X$ be the number of people whose coin and the coins of both neighbors land heads. a) Compute $\mathbf{...
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Seating in a cinema hall [on hold]

There are $N$ numebered seats in a cinema hall. Each seat has been assigned to a specific spectator. However, the first $N-1$ spectators arrive early and occupy the seats completely randomly. The last ...
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Expected Value of a recursive function

I want to calculate an expected Value for a given random Variable. I have $c_0 = 0,$ $c_{i+1} = \max \lbrace c_i,r_{i+1} \rbrace + d_{i+1},$ where $c_i,r_i,d_i$ are continuous random variables with ...
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The expected-value of the square of Sample Variance.

Suppose $X_1, \cdots, X_n$ are i.d.d. samples from population $X \sim N(\mu,\sigma^2)$, and the sample variance is denoted by $T = \sum_{i = 1}^n \frac{(X_i - \overline{X})^2}{n}$. I am curious about ...
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How to find $E(|X+Y|^3)$?

Supposed that $X,Y$ are independent random variables, and $X\sim N(\mu,\sigma^2),Y\sim N(-\mu,\sigma^2)$. I would like to culculate the value of $E(|X+Y|^3)$. I have thought of some methods but I was ...
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Finding Expectation of Continuous Joint Density

I am working on a problem and am a bit confused. The problem: Consider R.V. X and Y distributed on the triangle {(x,y) $\in$ $\mathbb R^2$| 0$\le$X$\le$1, 0$\le$Y$\le$x} p(x,y) = 2, 0$\le$X$\le$1, ...
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Finding Conditional Distributions

I am working on a problem and am a bit stuck. The problem: Find conditional distribution and expectation of X given Y (Sorry I did not know how to format into a table) \begin{array}{l:l|c}X & Y ...
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Solving for Conditional Variance

I am working on a problem and am a little bit confused. The problem: P(X=0,Y=0) = .80 P(X=1,Y=0) = .05 P(X=0,Y=1) = .025 P(X=1,Y=1) = .125 Find Var(Y|X=1) What I have done so far: (using) E(Y$^...
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Formula for finding Conditional Variance

I am working on a problem that asks for Var(Y|X=1) Is this the same as E(Y$^2$|X=1) - $\big($E(Y|X=1)$\big)$$^2$
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Expectation of Conditional Joint Density

I am working on a problem and am really confused. Problem: Find E(X|Y) with P(X,Y) = (1,1) = .4 P(X,Y) = (1,0) = .5 P(X,Y) = (0,0) = .1 What I have done so far: I drew a x y graph (plotting the ...
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If we throw a coin infinitely many times, counting $1$ for heads and $-1$ for tails, can we prove that we throw $x$ heads in a row in finite time?

We have the following setting; let $\{X_{i}:i\in\mathbb{N}\}$ be a sequence of independent identically distributed random variables representing the outcomes of our coin tosses, i.e. taking values $1$ ...
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What is the expected result for the number of heads obtained in this coin/dice flipping example?

Suppose you roll one fair six-sided die and then flip as many coins as the number showing on the die. (For example, if the die shows 4, then you flip four coins.) Let Y be the number of heads obtained....
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What is the meaning, both intuitively and mathematically, behind the probability of a set?

I'm currently studying a course on applied probability and I keep running into a notation that I can't quite understand. Let's say we are working in a progrability space $(\Omega,\mathcal{F},P)$ and ...
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Expected value of random variable which is the number of drawns required

An urn has n balls and we take balls from it with replacement until each ball has been drawn at least one. X denotes the number of draws required. What is the expected value and variance of X? I ...
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Normal order statistic. Expected Values (Fisher-Yates)

We have $\xi_{i} \sim \mathcal{N}(0,1)$ iid, $i= 1,\dots N$. We look for $a_i = \mathbb{E} \xi_{i:N} $, where $\xi_{i:N}$ is order statistic. I already checked in R, that ...
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Find the pdf of $Y=e^{X^2}$, where $X\sim N(0,1)$

$Y=e^{X^2}$ , where $X\sim N(0,1)$ I want to find the pdf of $Y$.
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Show $E\left[\left(\frac{\partial}{\partial\theta} \ln f(X)\right)^2\right]=-E\left[\frac{\partial^2}{\partial\theta^2} \ln f(X)\right]$

I encountered a question given $\theta$ in a random sample of size $n$ and also $$-n\times E\left[\frac{\partial^2}{\partial \theta^2}\times \ln f(X)\right]$$ , where $f(x)$ is the p.d.f. at $x$, ...
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Lower bound to expectation expression

Suppose I have a random variable $x$ and $E[x] =y$. Moreover $E[x^2] \leq M$. I have $E[\frac{x}{\sqrt{x^2 + c}}y]$ where $c$ is some positive constant. My question is can be show a (tight) lower ...
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Expectation of non-negative real-valued random variable

in some lecture notes, I stumbled upon the following formula : that if $X$ is a non-negative real-valued random variable then : $\mathbb{E}[X] = \sum_{n \geq0} \mathbb{P}(X \geq n)$ I tried ...