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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Equality in Gaussian Poincare Inequality

The Gaussian Poincare inequality states that: for $f: \mathbb{R}^n \to \mathbb{R}$ and $Z\sim \mathcal{N}(0,I)$, we have that \begin{align} Var(f(Z)) \le E[ \| \nabla f(Z)\|^2]. \end{align} My ...
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When mathematical statistics outsmarts probability theory

This is not a question, but it is too good to pass. I read it is originally due to Enis, Peter. "On the relation $E (X) = E [E (X∣ Y)]$." Biometrika 60, no. 2 (1973): 432-433. Assume $Y$ has ...
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Is it valid to move a logarithm inside of an expectation?

I have the following derivation for a latent variable model. $$ \newcommand{\d}[1]{\mathrm{d}#1} $$ $$ \begin{align*} \mathbb{E}_{q(x_0)}\left[-\log p_\theta(x_0)\right]\\ &=^{1.} \mathbb{E}_{q(...
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When is a probability a function of a probability?

I am asked to calculate $E(ln (\frac{1}{P(X)}))$ Where X is a discrete random variable over $\chi $ $\in \{1,2,3\}$ with probability mass function P(X=1)=0.5,P(X=2)=0.4, P(X=3) = 0.1 I think this ...
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$cov(X,f(X))\neq 0$ and $E(X f(X))\neq 0$

Take a random variable $X$. Is it true that (1) $cov(X,f(X))\neq 0$ for any function $f$? (2) $E(X f(X))\neq 0$ for any function $f$? I believe the answer to both questions is no. However, can you ...
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An Equivalence Statement of Two Expected Values

Suppose that we have three random variables, $V_1,\;V_2,\;V_3$. I derived an equivalence statement through a pretty long derivation: $$\mathbb{E}[V_1|V_2=v_2,\;c>V_3]=\mathbb{E}\left[\;\mathbb{E}\{...
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What is the relationship between the manifest correlation between ranked variables and the latent, continuous correlation?

Suppose you draw n pairs of observations from a real-valued bivariate distribution. You then convert each observation to its ascending rank order. Given the population correlation for the real-valued ...
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Why is the expectation of a random vector still a vector?

In my previous post on derivation of covariance between y and random effect, for the following linear model: Frank's anwer proved cov(y, u) = ZD as below: Frank's proof involved E[u]. As far as I ...
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Why do we need importance sampling?

Let's say we want to calculate the following expectation: $$ \mathbb{E}_{z\sim p_z(z)}[f(z)] $$ One issue, is that the samples from $p_z(z)$ could be not very informative: We see here that $f(z)$ ...
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What is the expected value of this process?

There are $n$ piles each containing $a_i$ stones. In a sequence of moves, Alex chooses two neighbouring piles randomly (containing, say, $A$ and $B$ stones) and combines them to create a single pile ...
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How to calculate an expected value of this expression?

I have three random variables: $X$, $Y$, and $Z$ and I know their expected values. I want to calculate: $$E\left[\frac{3-X-Y}{Z}\right].$$ I know that $E\left[\frac{1}{X}\right] \neq \frac{1}{E\left[ ...
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How to simulate the St. Petersburg paradox

I am trying to find the expected value of the game in the St. Petersburg paradox. The game has infinite expected value, but in my simulation the payouts are about 15 on average, which is way too small....
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Expectation of the log of a negative random variable

I read the following result based on a second order Taylor expansion (here: Expected value of a natural logarithm) $$\mathbb{E}(\log(x)) = \log(\mathbb{E}(x))-\frac{1}{2}\frac{\mathbb{V}\text{ar}(x)}{\...
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Expected value of max of $k$ linear combinations of Bernoulli random variables

Let $X\in\mathbb{R}^{n\times k}$ be a matrix with entries $x_{ij}$ and $(B_1,\dots,B_n)$ denote $n$ independent Bernoulli random variables with probability of success $p$, and furthermore let $Y=\max_{...
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Law of large numbers for transformed and non-transformed random variables

Law of large numbers states that: If $X_1, \dots X_n \sim p(x)$ are IID, then $ \frac{1}{n} \sum_{i=1}^{n} X_i \rightarrow \mathbb{E}_{p(x)}\{X\}= \mu$, where $X \sim p(x)$. Below is what I'm having ...
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sum of expected differences in time series

I have a (markov) decision process without reward. I am estimating the expected state differences $\mathbb{E}[\delta_t]$ where $\delta_t = X_{t+1} - X_t$. A state $X_{t+1}$ can be expressed as $X_0 + \...
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What is the sum $\sum_{m} e^{i (U_m k + \beta_m)} $ when $U$ and $\beta$ follow different distributions

I have the following function. $$ x(k) = \sum_{m} e^{i (U_m k + \beta_m)} $$ $i = \sqrt{-1}$ Here, $U_m$ are samples drawn from a Gaussian random distribution. $$ U_m \sim \mathcal{N}(\mu, \sigma) $$ ...
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Positive expected value for lottery

As far as I know, to decide either you should enter a bet or not, you should get the expected value of that bet I was wondering if the lottery has a very high expected value, is it wise to join? There ...
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Variance of $U= a \log (Z+b)-Z$ where $Z$ is the exponential random variable

Consider a random variable \begin{align} U= a \log (Z+b)-Z \end{align} where $a,b>0$ and $Z$ is an exponential random variable. Question: Can we find the variance of $U$? Things that I tried ...
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What is the expectation of the exponential distribution multiplied by indicator function?

I am reading the research paper [A New Bayesian Lasso], where $u$ has the distribution The expectation of $u_j$ is given by $$\frac{1}{\lambda}+|\beta_j|$$. I know that the term $\frac{1}{\lambda}$ ...
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Expectation of a Hebbian term

In a paper by Akrout et al., it is mentioned that [Given $\mathbf{y} = \mathbf{W}\mathbf{x}$, where $\mathbf{x}\in\mathbb{R}^m$ is an input vector, $\mathbf{W}\in\mathbb{R}^{m\times n}$ is a matrix, ...
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Expected value of a product of two ReLU functions

Let assume $\theta,\theta'\in R^d$ (are two fixed $d$ dimensional real-valued vectors) and $x\sim N(0,I_d)$ (is a d-dimensional random vector comes from standard normal distribution). Now, I am ...
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Is the expected value of a probability over an interval meaningful?

I am reading an unpublished manuscript and have come across an equation of the following form for the calculation of the probability of an even A, $$ P[A]=E\Big[P[X>x|Y]\Big]. \tag{1} \label{1} $$ ...
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100-sided dice roll problem

When should I stop rolling if it costs $1 for each roll and I earn only the value of the final roll shown on a 100-sided dice roll? My intent is to maximise profit and I have unlimited rolls
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Expected value of a complicated function

I want a closed form/ semi closed form of the expected value of a complicated function. The function looks like this, $$ f(x) = \frac{\sin(A \frac{x}{2})}{\sin(\frac{x}{2})} \frac{\sin(M B\frac{x}{2})}...
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Implication of $E(XY)\neq 0$

Consider two real-valued scalar random variables $X,Y$ such that $$ (1) \quad E(XY)\neq 0,\quad E(X)= 0, \quad E(Y)= 0 $$ Let $g: \mathbb{R}\rightarrow \mathbb{R}$ be a function. Under which ...
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Expected number of random cell visits in Q-learning

I'm interested in Q-learning with a time-decaying exploration rate $$ \varepsilon_t = \exp(-\beta t)$$ In order to evaluate how `reasonable' certain values of $\beta$ are, and to make them comparable ...
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Expected value of [variance of sample means] is the average of [sample mean variance]?

This is drawn from Gelman et al in Regression and Other Stories: If I take $k$ independent sample proportions $p_i$ with differing sample sizes $n_i$, Gelman et al says that the observed variance of ...
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Expectation of truncated distribution

Consider the random variables $X,Y$ and assume that $$ E(X|Y)=0 $$ Does this imply that $$E(X|X\geq A,Y)\neq 0 ?$$ I think this holds for the truncated Normal, for example. But does it hold ...
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In an RCT, does running OLS on $Y_i = \beta_0 + \tau D_i + \varepsilon_i$ and recovering $\tau$ recover ATE or ATT

Let's say I run an RCT and then run OLS on $Y_i = \beta_0 + \tau D_i + \varepsilon_i$ where $D_i$ is a dummy variable indicating whether an individual $i$ received the treatment. If I were to take the ...
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assessing which random sample agrees more with a preferred ranking

I am not a mathematician or a statistician. But, I think the question I have is related to statistics. I will start with a made up example. If I can grade apples into,say four grades from 1 to 4, one ...
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Average Error vs. Aggregate Error

I was reading this paper on the history of Bagging Estimators (https://www.stat.berkeley.edu/~breiman/bagging.pdf) and came across the following section: I am having difficulty understanding the ...
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Expectation of the product of multiple correlated 1-D normal variables [duplicate]

First question: is it possible to have a set of $k$ random variables $\left\{X_i\right\}$ s.t. each $X_i \sim N(0,1)$ individually, and $\text{Corr}(X_i,X_j)=\rho$, $∀i\neq j$? If those conditions are ...
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1 answer
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Sum of probabilities of all samples gives the total volume?

I am working on a churn problem (binary classification - whether a customer will churn or not). Now using logistic regression, I get the probability whether the customer will churn or not. Can I add ...
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h1b lottery procedure change to increase proportion of masters. Can someone explain how the expected proportion is different in the new procedure?

Simply speaking, there are 85,000 spots. 20,000 is reserved for masters(or higher education). Previously, they were selecting 20k first from everyone with a masters and then grouping everyone ...
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16 votes
3 answers
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Expectation of the product of iid random variables

If we have iid random variables $X_1,X_2,...,X_N$ with $\mathbb{E}X_i=\mu$, is it true that $\mathbb{E}\prod X_i=\mu^N$? I had no doubt that this is true, until I tried it out with Python, using ...
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1 vote
2 answers
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Expected remaining life with constant probability of death

In a famouse paper by Olivier Blanchard ("Debt, Deficits, and Finite Horizons" The Journal of Political Economy, Vol. 93, No. 2. (Apr., 1985), pp. 223-247.) he claims the following: "If ...
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Conditional Expectation of Random Variable given an event

Suppose $(\Omega, \mathcal{H}, \mathbb{P})$ is a probability space, $(\mathsf{E}, \mathcal{E})$ a measurable space and $X:\Omega\to \mathsf{E}$ a random variable with well-defined expectation $\mathbb{...
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Bayesian Quadrature of Expectation w.r.t. Kernel Density Estimator Probability Density

I have a model of a physical system, $f(\pmb{x})$, where $f$ is the output of a mathematical model and $\pmb{x}$ are inputs to the model, which are available as observations. My goal is to find the ...
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expected value of conditional uniform distribution

Suppose $U$ is the uniform distribution from $0$ to $1$. I wish to compute $E(U|U<1/2)$. Intuition suggests that this should just be $\frac{1}{4}$. However, when I try and compute it explicitly, I ...
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Proof of the Student t-test for independent samples drawn from the same normal distribution when $\mu \neq 0$

I'm following the proof in Cramer's book Mathematical Methods of Statistics, $\S 29.4$. There it is assumed that we have two independent samples $x_1,\ldots, x_{n_1}$ and $y_1,\ldots,y_{n_2}$ drawn ...
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1 answer
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Finding the maximum likelihood solution corresponds to finding the root of a regression function. How?

Given a pair of RVs $z,\theta$ governed by a joint distribution $p(z,\theta)$. Conditional expectation of $z$ given $\theta$ defines a deterministic function (called as regression functions) $f(\theta)...
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General Closed Form and dispersion parameter of the Expected Maximum of i.i.d Gumbel Variables

I would like to know the general closed form of the expected maximum of i.i.d Gumbel variables. I found this onlie: Expectation of the Maximum of iid Gumbel Variables In the linked page, it shows the ...
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Is there name for statistical thinking of -- say a basket of 60% red, 40% black balls --- NOT guessing optimally?

I remember reading about this. Say there's a basket of 60% red balls and 40% black balls. And many statistical laymen if say given a reward for a correct guess -- they will guess red and black and etc....
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How to show that simple random sample sensitivity is unbiased for population sensitivity

In diagnostic testing, sensitivity $S$ is the probability that the test gives a positive result given that you have the condition being tested. From a simple random sample of people who take the test, ...
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$E(SN)$ for aggregate claim amount $S$, $S=X_{1}+...+X_{n}, X_{i}$ are iid [duplicate]

Consider the following model for aggregate claim amounts $S$: $S=X_{1}+X_{2}+...+X_{N}$ where the $X_{i}$ are independent, identically distributed random variables representing individual claim ...
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In a time series $x_t, x_{t-1},...,$, why is $E[x_t|x_t, x_{t-1},...]= x_t$?

Sorry if obvious but in a time series $x_t, x_{t-1},...,$, why is $E[x_t|x_t, x_{t-1},...]= x_t$? I don't really get what the random variable $x_t|x_t, x_{t-1},...$ represents? What I find ...
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Calculating expected value

Came across an interesting problem. You’re clearing out your garage for a garage sale, and you want to get rid of as much stuff as possible quickly. You found a dresser and decided to sell it to the ...
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2 answers
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Notation of expectation with conditional in subscript

Inside the book "The Elements of statistical learning", I stumbled upon the following notation (Ex. 2.7) $$E_{\mathcal{Y|X}}(f(x_0) - \hat{f}(x_0))^2$$ where $\mathcal{X, Y}$ are two random ...
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Expected value of a random variable with truncation

Let $f:[0,\infty)\to \mathbb R_+$ denote the PDF of a random variable $X$ and $c>0$ a constant. I want to evaluate the following integral: $$I(c)=\int_0^\infty{\min(x,c)f(x)dx}.$$ This can be ...
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