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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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Second moment of weighted average of random variables

I stumbled upon problem 254 from the SOA Exam P list in https://www.soa.org/globalassets/assets/Files/Edu/edu-exam-p-sample-quest.pdf for which I am puzzled by the solution described in https://www....
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analytical asymptotic approximation of the expected maximum, mean, and minimum distance of nearest neighbours in unit ball

Say I uniformly at random distribute $x = n^3$ (independent identically distributed) points in a ball of radius $r=1$ in $\mathbb{R}^3$. What can be said about the expected maximum, minimum, and mean ...
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how can predictive distributions be considered as expectations?

I guess that the prior and posterior predictive distributions can be considered expectation of $p(y|\theta )$ (in case of prior predictive distribution) and $p(\widetilde{y}|\theta )$ (in case of ...
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Expectation of binomial random variable

Having trouble understanding something I read in a paper recently. Say we have $X \sim \operatorname{Binomial}(N,p).$ The paper states: $$E[X \mid N,p] = Np$$ (so far so good) and $$E[X] = \mu p$$ ...
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Convex predictive mean of Gaussian Process

In Gaussian process (GP) regression, predictive mean is $$ K(X^*,X)[K(X,X)+\sigma^2I]^{-1} \textbf{y}$$ Is there a method to ensure that the predictive mean is convex with respect to the test input $X^...
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I don't think this is conditional dependence, so what is it?

I am looking for the name of the following phenomenon. There are three random variables, $X,Y,Z$. We have $P(X,Y) \neq P(X)P(Y)$ and $P(Y,Z) \neq P(Y)P(Z)$. In other words, $X$ and $Y$ are dependent, ...
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I would like some insight into what I have been working on here

I work in roofing sales which involves door-knocking at the entry level. Our daily numbers are printed in our GroupMe chat for our branch. I took it upon myself to do some analysis on those numbers. ...
therealchriswoodward's user avatar
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Expectation under convex order by multiplying

I am trying to understand if the following statement is true, or the conditions under it is satisfied. Let $M,N$ and $X>0$ be random variables. If the following inequality holds for any concave non-...
Don P.'s user avatar
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Computing a Confidence Interval for E[X] when PMF is given

I am given a Probability Mass Function for a discrete random variable. From the PMF I computed the Expected Value $E[X]$, the Variance $V[X]$ and the Standard Deviation $S[X]$. Here is an example (the ...
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Expected Value Chi Square distribution

I'm trying to simulate the distribution from the sample variance $s^2$ and compare it with the theoretical distribution. Therefore, I perform a fairly simple simulation (upfront, I'm not a ...
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Expectation under convex order

I am trying to understand if the following statement is true. Let $M,N$ and $X$ be random variables. If the following inequality holds for any concave non-decreasing function $u$ \begin{equation} \...
Don P.'s user avatar
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Distribution of outcomes of multiple binomial distributions

I have a sample of 50 subjects, where every subject completes a task with two possible outcomes (left or right hand use, with 50% probability) 30 times. On an individual level, this leads to a ...
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Expected value of a decreasing function of two random variables

My question is exactly equal to the question posted at Expected value of decreasing function of random variable versus expected value of random variable with just one extra assumption: the two random ...
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Bayesian Fubini Tonelli

I am working on a bayesian framework where I place a Gaussian Process on my function $f\sim GP$ and have data $D^n=\{(X_i,Z_i,W_i)\}^n$. I then have the posterior measure $\mu(f|D^n)$. The posterior ...
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Validating binary prediction model

Suppose we have a model that predicts for binary event $e$ ($0$ or $1$) with a single output $p$ (the expected probability $e$ occurs). If we are able to compare $p$ with the true value of $e$ ($0$ or ...
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Deriving home vs away goals - using total expected goals and home/draw/away probabilities

In the context of a football ("soccer") match, if I have the following for a single game: Probability of Team A winning Probability of Team B winning Probability of a draw The total goals ...
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$E[(X+Y)^{a}] > E[(X)^{a}]$?

Assume I have two strictly positive i.i.d. random variables, $X$ and $Y$. Under what conditions is the following inequality true? $$E[(X+Y)^{a}] > E[(X)^{a}], \hspace{2mm} a \in (0,1)$$ Should have ...
econ_ugrad's user avatar
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1 answer
275 views

Probability algorithm on strings

Let $x$ be any binary string $\in (0,1)^*.$ The majority language is given by: $$\text{MAJ}:=\{x\in (0,1)^*:\sum_{i=1}^ {|x|}x_i>\frac{|x|}{2}\},\text{where $x_i$ is the $i$-th position value(...
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Need help in calculating $\mathbb{E}(\frac{1}{x_{(2)}-x_{(1)}}\int_{x_{(1)}}^{x_{(2)}} f(t) \ dt)$, where $x_{(i)}$ are related Beta distribution

Suppose $Y, Z \stackrel{\text{iid}}{\sim}\mathrm{Uniform}(0,1)$. Let $a = g(\min(y,z)),\ b=g(\max(y,z)).$ How can I calculate the expectation $$\mathbb{E}\left[\frac{1}{b-a}\int_a^b f(t) \ dt\right]$$ ...
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Derive the expectation and variance of squared sample correlation: delta-method or else?

I would like to obtain the expectation and variance of the squared Pearson sample correlation ($\operatorname{E}(R_{lk}^2)$ and $V(R_{lk}^2)$) between two random variables $l$ and $k$ following a ...
CafféSospeso's user avatar
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Why does $E(V_n/(n+2)-1)^2=2/(n+2)$ when $V_n\sim\chi^2(n)$?

I was reading some lecture notes when I saw a simplification I didn't understand. Specifically, we have $V_n\sim\chi^2(n)$. It was then written then $$E\left(\frac{1}{n+2}V_n-1\right)^2=\frac{2}{n+2}.$...
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Expectation of $u^\top(u+Ax)$, when $A$ and $u$ are nonlinear functions of $x$

Let $x\in\mathbb R^d$, and $s=\operatorname{softmax}(x)$. Let $y$ be a fixed one-hot vector such that $$u = s-y \\ v =(\operatorname{diag}(s) - ss^\top)x$$ I am interested in the inequality $u^\top (u ...
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Expected value of Cosinus in High dimension

I would like to prove that the cosinus of the angle formed by 3 randomly points tends to $\frac{1}{2}$ as the dimensionality tends to $\infty$. Could it be solved with the expected value formula ? It ...
Jérémy's user avatar
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Expectation over cost-normalized Expected improvements

Are the following two expressions equivalent if we assume the independence of f(x) and C(x)? $$ E\left[\frac{E\left[\max\left(f(x) - f(x^*), 0\right)\right]} {C(x)}\right] $$ $$ \frac{E\left[\max\...
Ridwan Salahuddeen's user avatar
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33 views

Expectation & Covariance matrix of indicator vector

Suppose we have the $p$-dimensional random vector $\boldsymbol{X} \sim \mathcal{N}(\boldsymbol{\mu}, \Sigma)$. Take the set $A$ to be (without loss of generality) the negative real line, thus $A = (- \...
HeyCool08's user avatar
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How to calculate the expectation of the following Dirichlet distribution and Beta distribution?

This is a question from my research, related to the derivation of the variational EM algorithm with mean-field assumption about LDA-based model. We all know, given that $\boldsymbol{\theta} \sim \...
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Verifying the integrability condition of a deterministic volatility function

Suppose there is integrability condition: \begin{equation} \mathbb{E}\left[\int_0^T\frac{\sigma^2(t)}{T-t}dt\right]<\infty \end{equation} for an arbitrary volatility function. Suppose I nominate ...
Carl's user avatar
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How to derive the expectation of $\log[a \theta_k + b]$ in Dirichlet distribution?

Given that $\boldsymbol{\theta} \sim \mathrm{Dir}(\boldsymbol{\alpha})$, then $E_{p(\boldsymbol{\theta} \mid \boldsymbol{\alpha})}[\log{\theta_k}] = \Psi(\alpha_k) - \Psi(\sum_{k'=1}^K \alpha_{k'})$, ...
Henry Zha's user avatar
2 votes
1 answer
132 views

Expected value of the product of three random variables

For two dependent random variables we have: $$Cov[X, Y] = E[XY] - E[X]E[Y]$$ So that $E[XY] = E[X]E[Y] + Cov[X, Y]$ In case of three arbitrarily correlated random variables $(X, Y, Z)$, is it possible ...
Stefano Lombardi's user avatar
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5 answers
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Is a product that has 4.9 stars from ten customers better than one that has 4.5 stars from a hundred customers?

In many areas, we encounter a situation where we compare averages of highly skewed statistics using two unequally sized samples. Typically, this happens when comparing items in an online store. For ...
2 votes
1 answer
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Tight upper bound on the function of expected value

Let $R$ be a positive integer, $\mathcal{X}$ be the sample space and $x \in \mathcal{X}$ be an event of the sample space; $P(x)$ denotes the probability of occurrence of event $x$. The problem is to ...
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What conditions are there on the exponent $p$ such that $\underset{\mu}{\arg\min}\left\{\mathbb E\left\vert X-\mu\right\vert^p\right\} $ must exist?

Let $X\sim F(x)$ be a (univariate) random variable defined by distribution function $F$. If the expected value exists, it is equal to $ \mathbb E[X] = \underset{\mu}{\arg\min}\left\{\mathbb E\left\...
Dave's user avatar
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Modelling Y=min(c,X) for different c

Assume I have a random variable $X \sim Poisson(\lambda)$ which models the potential nr of people entering some room. Now consider this room has a capacity $c$ so that whenever $X > c$ we observe $...
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Find $E[Y]$ when $f(x,y) = \frac{x}{3}e^{-xy}$

Truth be told, I don't really have an issue with this problem in general, but in it's calculation. Let me explain. We need to find $E[Y]$ when $f(x,y) = \frac{x}{3}e^{-xy}$, $1<x<4$ and $y>0$...
Anweshan Goswami's user avatar
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What is the bias of uniform distribution parameter estimator?

I have a question regarding question 2 of chapter 6 of "All of Statistics" book by Larry Wasserman. let: $$X_1, ... , X_n \sim \operatorname{Uniform}(0, \theta )$$ and let: $$\hat{\theta} = \...
George Wilhelm Hegel's user avatar
5 votes
2 answers
288 views

Expected value of decreasing function of random variable versus expected value of random variable

Given two random variables $X_1$ and $X_2$ (same sample space $\mathcal{X}$) that $$\mathbb{E}[X_1]=\int_{\mathcal{X}}xf_1(x)dx > \mathbb{E}[X_2]=\int_{\mathcal{X}}x f_2(x)dx$$ Can we say that $\...
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How to calculate the expectancy of the ratio of non-independent random variables?

How can I calculate this expectancy: $$ E \left [ \frac{\sum_{t=1}^T{Z_tX_t}}{\sum_{t=1}^T{Z_t^2}} \right ] $$ where $Z_t \sim N(0,1)$ and $X_t \sim N(0,1)$ are independent? Any tricks? Is it ...
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Example in which $E[E[Y|T,X]] \neq E[Y|T, E[X]]$

Context: this question is a follow-up of this other question in which I'm trying to understand why we should use methods for causal inference instead of just training machine learning regressors, ...
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1 answer
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properties of a expectation for a non-negative random variable

Say I have a non-negative discrete random variable $X$ (values of $X$ can be mapped to integers $(0, 2^n -1)$ for $n \in \mathbb{Z}$) and an associated distribution $P(X)$. Given a non-negative scalar ...
Manas Sajjan's user avatar
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Correct notation for uncertain expectation

I need to write some documentation for a couple of process design options. Ultimately, my organization has to find a way to estimate the values of vector $A$. I have come up with a model to calculate $...
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Expectation of KL-divergence only as the log ratio of the probabilities

In the DPO paper, and in particular in the proof attached below, how can we expand the KL divergence only as the log ratio of the probabilities of the two distributions? According to the definition ...
goodolddays's user avatar
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How to find probability from $E[X^n]$?

It is given that $E[X^n] = \frac{2}{5}(-1)^n + \frac{2^{n+1}}{5}+\frac{1}{5}$, where $n=1,2,3,\ldots.$ I need to find $P(|X-\frac{1}{2}| > 1)$. What my approach is : I have opened the modulus ...
Anweshan Goswami's user avatar
2 votes
0 answers
93 views

$E(XY)$ for a truncated bivariate normal

If $(X, Y)$ follows a bivariate Gaussian distribution with mean ${\bf \mu}$ and covariance ${\bf \Sigma}$ with truncation bounds $(a_x, b_x, a_y, b_y)$, can we compute $E(XY)$ in closed form? If not, ...
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Expected average distance in greedy matching on a circle

Now we have several independent and identically distributed random variables following the uniform distribution on the interval [0, 1].They are denoted as $x_1, x_2, x_3, ..., x_m$ and $y_1, y_2, ..., ...
user11850847's user avatar
2 votes
3 answers
105 views

Scaling the conditioned random variable does not change conditional distribution, why?

Given two random variables $X$ and $Y$, I know intuitively that $$ \mathbb{E}[X\,|\,Y]=\mathbb{E}[X\,|\,cY], $$ where $c$ is some non-random constant. My intuition tells me that scaling the ...
Resu's user avatar
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Expectation of Mahalanobis Distance and its logarithm

Suppose: \begin{equation} X \sim \mathcal{N}(X, \mu, \Sigma_x) \text{ st. } \Sigma_x \sim \mathcal{IW}(\Sigma_x; \Psi, v) \end{equation} Where $\mathcal{IW}$ is the Inverse-Wishart distribution. This ...
Snowy Baboon's user avatar
2 votes
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Describing guaranteed profit situations which are stronger than just 'superfair wager'

Context. I am tutoring a final year secondary school student in mathematics. To illustrate the principles of card-counting in a situation of sampling without replacement, I've decided to show her a ...
microhaus's user avatar
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10 votes
2 answers
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Expected value of the square root of a lognormal variable

Let $X$ be a positive, lognormal random variable with known mean $\mu_X$ and variance $\sigma_X^2$. Since $X$ is a lognormal random variable, I know its pdf and moment-generating function (mgf). pdf: $...
LJ Beinhauer's user avatar
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Question on the proof step in the theorem 1 of the Gap statistic paper

From the Gap statistic paper, during the proof for the theorem 1, we can see the below equality (p. 422), $\begin{aligned} \operatorname{var}(X) & =\frac{1}{2} \int_{-\infty}^{\infty} \int_{-\...
kurtkim's user avatar
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When the expected value of the gradient of function is equal to the expected value of the function multiplied by the gradient itself?

I'm having doubts on the conditions of an equality. Considering $f(X,w): \mathbb{R}^n \xrightarrow{} \mathbb{R}^n$ a function of n-variate random variable $X$ with an unknown distribution $p(x)$ and $...
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