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Questions tagged [expected-value]

The expected value of a random variable is a weighted average of all possible values a random variable can take on, with the weights equal to the probability of taking on that value.

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Find $E[(X+Y)^2]$ given $E(X^2)=3$ $E(Y^2)=4$ and $E(XY) = 2$

I know this is an easy question but I'm having problems solving it I sorta thought that you'd get $E(X^2) + 2E(XY) + E(Y^2)$ and that'd add up to: $3+4+4=11$ but my answer isn't correct. I'm guessing ...
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1answer
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Does this expectation inequality holds?

Let $X\in L_p(P), p>1$. Is the following result true? $$E[\lvert X\rvert I(\lvert X\rvert>C)]\leq C^{1-p}E\lvert X\rvert^p.$$ where $C>0$. It can be found in the proof of Corollary A.1 (...
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Expected value for a function with a normal distributed random variable

I have a random variable $X \sim N(\mu, \sigma^2)$ and a function $5x^2 + 2x$. How can I calculate $E(g(x))$ ? I have two ideas, altough I'm not sure which one is right: $E(g(x) = \int_{-\infty}^{\...
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20 views

Prove stationarity

It can be easy question, but it is a part of bigger exercise. I have problem with lack of statement "uncorrelated" Prove that if ${X_t,t \in T}$ is stationary, then $Y_t = X_t - X_{t-1}$ is also ...
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Conditional expectation of an exponential functional [on hold]

Consider the function $g(\tilde{w})=-e^{-\delta\tilde{w}}$, where $\delta>0$ some constant and $\tilde{w}$ is some random variable s.t.$\tilde{w}=\alpha-\beta\tilde{x}+\tilde{y}\tilde{z}$. ...
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1answer
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How to compute $E[ (|X|) X]$ when $X \sim N(0,1)$?

Any help would be appreciated.
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Why do we use the Greek letter μ (Mu) to denote population mean or expected value in probability and statistics

According to this Wikipedia entry, "Mu was derived from the Egyptian hieroglyphic symbol for water, which had been simplified by the Phoenicians and named after their word for water". So, my question ...
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Maximum of an iid Gumbel Distribution for a Second Choice term (conditional on the first choice)

I am trying to understand the extension of Small and Rosen's (1981) closed form solution for compensating variation, as found in Fan (2013). In Fan's model consumers are choosing to subscribe to ...
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56 views

Prove $E[f(x)|X]=f(x)$ [closed]

I know: $E[f(x)|X]$ $=\int f(x)* \frac{f(f(x),x)}{f_x(x)} dx$ $=\int f(f(x),x) dx$ then I am not sure what to do next. If I take the integration of a pdf, then I will get the cdf of $(f(x),x)$ ? ...
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2answers
40 views

Conditional Expectation and Prediction

This is taken from the book "A First Course in Probability" from Sheldon Ross: Sometimes a situation arises in which the value of a random variable $X$ is observed and then, on the basis of the ...
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Finding the maximum and minimum variance of the sum of two Bernoulli events?

You are guessing the contents of two envelopes. Let $U_i$ be the event that you guess correctly. Your probability of guessing correctly for each envelope is $P(U_1) = P(U_2) = 3/4$. $U_1$ and $U_2$ ...
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Probability that the expectation lies in an interval

I am trying to solve the first one of these exercises: The zero-one loss is 1 if a sample is misclassified and 0 if it classified correctly. The empirical risk is the sample mean of the loss. The ...
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1answer
35 views

Finding expected values from joint distribution

For my textbook, Introduction to Probability by Blitzstein and Hwang, I have the problem where I have the random variables $X$, $Y$, and $Z$ such that $X \sim N(0, 1)$. I am also told that, ...
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Expected value for f(x) [closed]

I am reading an article and trying to extend their case to a multivariate case. I have the function $f_{i} (x)=\frac{1}{|Σ|}f_{0}((x-μ_{i})'Σ^{-1}(x-μ_{i}))$, where $f_{0}(.)$ is a base density ...
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Suppose $\frac{1}{Y} \sim Exponential(\lambda)$, what is the pdf of Y? [closed]

I am actually attempting to answer the following: Suppose $\Pr(X|Y=y) \sim \Gamma(a,y)$ and that $\frac{1}{Y} \sim Exponential(\lambda)$ What is $\mathbb{E}(X)$? However, I know that this is a ...
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Help on expectation of squares of sums [closed]

How does one do this: $E[(\sum X_i)^2]$, given that the Xi are i.i.d.?
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Why is the expected gradient of a density not parallel to the expected gradient of the log density?

I'm confused by a seemingly counter-intuitive property of the interaction between distributions, log transforms, expectations and gradients. Suppose I have some distribution over random variable $x$ ...
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Finding the expected value of Laplace distribution for specific interval

Hello I am experiencing difficulties finding the solution for the following Problem: Find the expectation of X where X has a Laplace distribution on [5, 9]. I understand how to find the expectation ...
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Statistical error in the approximation-estimation tradeoff

Show that $$E(g_\tau ^G(X)-g^* (X))^2 = E(X^T \hat{\beta}-X^T\beta^G)^2+E(X^T\beta^G-g^*(X))^2$$ where $g_\tau ^G(X) = X^T \hat{\beta}$ and $g^G(X) = X^T \beta^G$ where G is a class of linear ...
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Expected Optimism 0-1 Loss with 0-1 Response

Want to show that $$ E_X op = \frac{2}{n} \sum_{i=1}^n Cov_X(g(x_i), Y_i)$$ For 0-1 loss function with 0-1 response. Want I've done $$op = l_{in}(g) - l(g)=\frac{1}{n}\sum_{i=1} ^n Loss(Y_i', g(...
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What‘s wrong with my proof of the Law of Total Variance?

According to the Law of Total Variance, $$\operatorname{Var}(X)=\operatorname{E}(\operatorname{Var}(X\mid Y)) + \operatorname{Var}(\operatorname{E}(X\mid Y))$$ When trying to prove it, I write $$ \...
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Payout from a coin-flipping game with a bound on maximum payout

Suppose I play a game in which I flip a fair coin till I get tails. Let $N$ be the number of heads in the sequence of tosses. If I get a payoff of $\min(2^N, X)$, where $X$ is a constant like, say, $\$...
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Is the sample quantile unbiased for the true quantile?

I would like to find a way to show whether the sample quantile is an unbiased estimator of the true quantiles. Let $F$ be strictly increasing with density function $f$. I will define the $p$-th ...
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A maximization problem involving random variables: a special case

Consider random variables $X$ and $Y$ that are jointly normally distributed, $$ \begin{pmatrix} X \\ Y \end{pmatrix} \sim \mathcal{N} \left[ \begin{pmatrix} \color{blue}0 \\ \mu_Y \end{pmatrix} , \...
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understanding bias correction in ADAM algorithm [duplicate]

why is possible to approximate E[gi^2] with E[gt^2]? by the time we go to the t timestamp, we've already made weight updates, which mean gradients should be different as they are taken from different ...
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Deriving posterior mean with horseshoe prior

I want to decompose a matrix $S \in \mathbb{R}^{D \times D}$ as below $$S=vv^T $$ where $v_i\mid\lambda_i,\tau_i \sim N(0,\lambda^2_i\rho^2_i)$, $\lambda_i \sim Cauchy^+(0,1)$ i.e $v$ has horseshoe ...
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Does it make sense that the expectation of a random variable depends on the mapping?

For example The random process defined by "color observed in a pixel" have the following outcomes: $G, B, R$ And we have a discrete random variable $X_1$, which can create a mapping between each ...
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Law of total covariance with multiple conditionals

The law of total covariance states: $Cov(X,Y) = E(Cov(X,Y|Z)) + Cov(E(X|Z),E(Y|Z))$ If I condition on another variable $T$, does this still hold? E.g. $Cov(X,Y) = E(Cov(X,Y|Z,T)) + Cov(E(X|Z,T),E(Y|...
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Expected standard deviation of probabilistic sample

Let's say I have a collection of $n$ individuals, each individual is associated with a single value $x_i$, with $i = 1, 2, 3, ... n$. I now gather a sample of $s$ individuals from this collection, ...
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19 views

A maximization problem involving random variables

Consider random variables $X$ and $Y$ that are jointly normally distributed, $$ \begin{pmatrix} X \\ Y \end{pmatrix} \sim \mathcal{N} \left[ \begin{pmatrix} \mu_X \\ \mu_Y \end{pmatrix} , \begin{...
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1answer
25 views

The expected log-Likelihood in Kullback Leibler Divergence

Given a true normal distribution $g(x)$ with mean $\mu_G$ and variance $\sigma_G$, and a model $f(x)$, the KL divergence involves the expected log-likelihood $\mathbb{E}_G[log f(x|\theta]$. The ...
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5answers
120 views

How do I find all even moments (and odd moments) for $f_X(x)=\frac{1}{2}e^{-|x|}$?

I was asked to find a formula for all even moments of the form $E(X^{2n})$ and all odd moments of the form $E(X^{2n+1})$ using a mgf. Can you help me find the even moments? I will attempt to solve for ...
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0answers
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Joint expectation subscript when we only have single random variable

As a follow-up to these two questions, Subscript notation in expectations and Double expectation (Not law of iterated expectation) If we have $E_{XY}[X]$, what is the correct integral expression? i....
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+500

The frog problem with negative steps

Standard Problem description In this question The Frog Problem (puzzle in YouTube video) a frog has to jump from leaf to leaf on a row of leaves. And the question is how long it takes on average to ...
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2answers
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Intuitive Understanding of Expected Improvement for Gaussian Process

So I am learning Bayesian Optimization and came across expected improvement. My question is are we searching for the point in the Gaussian Process model whose expected value (determined by mean and ...
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14 views

Expected value for a sequence of group samplings from a fixed population

Suppose I've 100 apples where 25 of them are bad and the remaining 75 are good. I draw apples 20-by-20 from this group of apples. That is, I draw the first set of 20 apples from 100 apples, second ...
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4answers
821 views

Expected value until a success?

Suppose we have the a game with a 5-sided unfair die (just to make the probabilities easier to sum to 1), each side having a different payout. For each side $ x \in \{1,2,3,4\}$ we have the ...
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1answer
35 views

Problem Deriving Expected Sufficient Statistic

Any exponential family distribution can be expressed as $p(x|\theta) = g(\theta) f(x) e^{\phi(\theta)^T T(x)} = f(x) e^{\phi(\theta)^T T(x) - A(\theta)}$ where $A(\theta) = -\log{g(\theta)}$. We ...
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51 views

Expectation of a constant (basic topic)

Consider the space $(\Omega,\mathcal{F},P)$. Show that $E(c)=c,\forall c\in \mathbb{R}$. I thought about two distinct ways to show that. $E(c)=\int_\Omega c\ dP=c\int_\Omega dP=cP(\Omega)=c$; Let $f=...
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1answer
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autoregressive time series issue understanding expectation

Consider the following question How is it obvious that $\mathbb{E}(X_t) = 0$? Is it because of the following? Recursively we find that $\mathbb{E}(X_t) = \phi \mathbb{E}(X_{t-1}) = \phi^2 \mathbb{E}(...
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Expected value of ratio of random variables

Let X and Y be independent random variables with $$E(X) = 0\ and\ Y > 0$$ Find the mean value of $$ X/Y$$ My attempt: We have for independent random variables $$E(XY)=E(X)\times E(Y)$$ Hence, $...
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1answer
30 views

Chi-square Goodness of fit with specific expected values

Let's say I have the following summary data/observed counts. ...
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0answers
21 views

Expectation of Log Likelihood Function with given parameters Proof

I have been looking for AICc value derivation employing Kullback-Leibler distance but as a result of my search I got stuck with expectation of loglikelihood. In the link loglikelihood is given as $InL(...
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0answers
34 views

Express the mean acceptance rate of the Metropolis-Hastings algorithm as a total variation distance

Let $(E,\mathcal E,\lambda)$ be a measure space $p:E\to[0,\infty)$ be $\mathcal E$-measurable with $$c:=\int p\:{\rm d}\lambda\in(0,\infty)$$ and $$\mu:=\underbrace{\frac1cp}_{=:\:\tilde p}\lambda$$ $...
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1answer
148 views

Expression for the mean acceptance rate of the Metropolis-Hastings algorithm

Let $(E,\mathcal E,\lambda)$ be a measure space $p:E\to[0,\infty)$ be $\mathcal E$-measurable with $$c:=\int p\:{\rm d}\lambda\in(0,\infty)$$ and $$\mu:=\underbrace{\frac1cp}_{=:\:\tilde p}\lambda$$ $...
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0answers
22 views

L sets and Probability theory

I am required to prove the following: Let $L_1$ be the space of real-valued random variables on $(\Omega,\mathscr{A},\mathbb{P})$ which have finite expectation $\mathbb{E}(|X|)<\infty$ and let $...
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1answer
18 views

Expectation over complex multiplication is multiplication of expectation

Suppose I have $\mathbb{E}\left[{\bf x} {\bf x}^H \right]$, where ${\bf x} \in \mathbb{C}^{N \times 1}$ is a random vector which has a uniform distribution, then can I say, $\mathbb{E}\left[{\bf x} {\...
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1answer
32 views

Importance Sampling derivation

I'm learning about importance sampling from p139 of this book which has the following derviation: What I am confused about is the second step in the derivation, though the rest makes sense to me. I ...
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1answer
91 views

How to find Expectation?

If $X \sim N(0, \sigma^2)$ and $Y \sim N(0, \sigma^2)$ are independent, how can we find the expectation $$E \left(\frac{X }{\sqrt{X^2+Y^2} }\right)\,?$$
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0answers
13 views

Law of Total Expectation Estimation With Countably Infinite Support

I am attempting to estimate $E[W] = E[E[W | A]] = \sum E[W | a] \Pr (a)$. I have expressions for both the conditional expectation and the probability of a, where the probability is Poisson distributed ...