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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Expectation over cost-normalized Expected improvements
Are the following two expressions equivalent if we assume the independence of f(x) and C(x)?
$$
E\left[\frac{E\left[\max\left(f(x) - f(x^*), 0\right)\right]} {C(x)}\right]
$$
$$
\frac{E\left[\max\...
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Expectation & Covariance matrix of indicator vector
Suppose we have the $p$-dimensional random vector $\boldsymbol{X} \sim \mathcal{N}(\boldsymbol{\mu}, \Sigma)$. Take the set $A$ to be (without loss of generality) the negative real line, thus $A = (- \...
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How to calculate the expectation of the following Dirichlet distribution and Beta distribution?
This is a question from my research, related to the derivation of the variational EM algorithm with mean-field assumption about LDA-based model.
We all know, given that $\boldsymbol{\theta} \sim \...
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Verifying the integrability condition of a deterministic volatility function
Suppose there is integrability condition:
\begin{equation}
\mathbb{E}\left[\int_0^T\frac{\sigma^2(t)}{T-t}dt\right]<\infty
\end{equation}
for an arbitrary volatility function. Suppose I nominate ...
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How to derive the expectation of $\log[a \theta_k + b]$ in Dirichlet distribution?
Given that $\boldsymbol{\theta} \sim \mathrm{Dir}(\boldsymbol{\alpha})$, then $E_{p(\boldsymbol{\theta} \mid \boldsymbol{\alpha})}[\log{\theta_k}] = \Psi(\alpha_k) - \Psi(\sum_{k'=1}^K \alpha_{k'})$, ...
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Expected value of the product of three random variables
For two dependent random variables we have:
$$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 ...
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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 ...
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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\...
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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$...
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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} = \...
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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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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 ...
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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 ...
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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 ...
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$E(XY)$ 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, ..., ...
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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 ...
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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 ...
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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 ...
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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:
$...
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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_{-\...
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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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Gambling in multiple rounds with a maximum permitted bankroll and favorable or unfavourable probabilities
This is based on a deleted question, with the premises clarified to my understanding.
You are gambling in a casino with particular rules:
Bets are paid off at even amounts, so if you win a round you ...
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Calculating Expectation using the Linearity of Expectation law and sum of indicator random variables
I'm attempting to complete the problem sets for the Stanford CS109 Statistics course from 2021 as I follow along with the lectures.
I'm stuck on a particular problem in one of these problem sets. I ...
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Expectation and variance of bivariate skew normal distribution
I am fitting a bivariate skew normal distribution to a 2D data through the sn package in R. I get a $2 \times 1$ vector of ...
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Guessing the size of a set based on number of repeated random draws
I am trying to study a problem in algebraic number theory through a set of computational experiments. I have an enormous (say, of size $X$) family $\mathcal{F}$ of polynomials and I'm trying to ...
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Mean Squared Error for point estimation
I am attempting to understand Mean Squared Error when evaluating point estimators for particular parameters of interest. The book we are reading for class states the following:
The mean squared error (...
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Interpretation of expected wealth in Kelly betting paper
This is a sub-question of another StackOverflow question.
Kelly betting on horse races with uncertainty
in probability estimates (Metel 2017) describes an "ECC" variant of the Kelly method ...
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Using Law of iterated expectations, I want to calculate mean of Y, E(Y)
I obtain insights into calculating the conditional mean and
variance of Y given X, denoted as E(Y|X) and Var(Y|X) respectively. Building upon this
knowledge, I want to answer the follow-up question ...
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Ratio of Normal Distributions [duplicate]
Suppose I have two independent random variables, $X \sim N(\mu_1,\sigma_1^2)$ and $Y \sim N(\mu_2,\sigma_2^2)$ with $\mu_1,\mu_2 > 0$.
How can I compute/estimate
$$ \mathbb{E}\left[\left\lvert \...
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What are the expected residual standard deviations from each of the fitted models and data-generating process?
I simulate data to be analyzed using a linear mixed-effects model. It is based on an experiment with 2 levels (A and B) of a ...
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Expected value of largest eigen value of sample correlation matrix
Suppose $X$ follows some multivariate distribution with zero mean and Identity covariance matrix. Suppose $X$ is N dimensional. Suppose $R$ is the sample correlation matrix, calculated based on n ...
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Expectation of residuals in linear regression
Consider a linear regression model
$$ Y=X\beta + \epsilon, $$
where $Y\in R^n$, $X = (x_1,...,x_n)^T\in R^{n\times p}$ are i.i.d. $p$-dimensional observations, $\beta\in R^p$, and $\epsilon = (\...
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In this RL problem, why is the substitution $q_*(A_t)=\mathbb{E}[R_t | A_t] \to R_t $ valid within this expectation (over actions)?
The question that follows is from a machine learning textbook (Reinforcement learning Suttion and Barto page 39 link).
Given:
a probability distribution over actions $x$ (a policy) at time $t$ ...
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The training error of best hypothesis
Let $\mathcal{X}$ and $\mathcal{Y}$ denote the domain set and label set respectively. Also let $\mathcal{D}$ be a distribution over $\mathcal{X}$ and $f:\mathcal{X} \to \mathcal{Y}$ be the true ...
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$E[W\otimes W]$ for Wishart R.V. $W$
What is the value of $E[W\otimes W]$ for Wishart R.V. $W$?
$\otimes$ refers to Kronecker product
I found related formula for $E[WAW]$ on page 467 of Seber's Matrix handbook, wondering if $E[W\otimes W]...
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Confused on Kullback-Leibler divergence being invoked without proper definition
I am trying to understand how authors of the DDPM paper in appendix A, made the leap from equation 21 to equation 22.
Specifically, it is not clear to me how they managed to convert the first term of ...
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Estimate number of bad actors
This might be a silly question, but it's got me stumped. I am trying to estimate the number of bad actors in a system.
Let's suppose we have 100 users, and some percentage of them are bad actors. In ...
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How to find $\mathbb{E} \left[\frac{\bar{\mu}}{\bar{\sigma}^2}\right]$?
I asked the same question on math stacks: MathStacks:, and some user suggest to ask it here for better insight. So this question has found interest in many research problems, but there have been no ...
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Question regarding probability and maximum possible variance
I have the following question:
Suppose we have a set of 10 numbers (1, 2, ... , 10), each with a certain probability tagged to it.
Is it true that the highest possible variance is achieved when 1 and ...
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Joint density of two functions of a uniformly distributed random variable
I'd like to work out $\operatorname{Cov}(\cos(2U), \cos(3U))$ where $U$ is uniformly distributed on $[0, \pi]$.
I believe this involves computing $\mathbb{E}[\cos(2U)\cos(3U)]$. If so, then I first ...
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Cross-Validation estimate for the risk is almost unbiased
Let
$Z_N$ : set with N elements; full training set
$Z^l_{N/L}$ : set with N/L elements; l-th hold-out set
$Z_{N(1-1/L)}$ : set with N-N/L elements; e.g. 4/5 of data
$Z_N \setminus Z^l_{N/L}$ : ...
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Derive Cramer-Rao lower bound for $Var(\hat{\theta})$ given that $\mathbb{E}[\hat{\theta}U]=1$
I am trying to derive the Cramer-Rao lower bound for $Var(\hat{\theta})$ given that we already know $\mathbb{E}[U]=0$, $Var(U)=I(\theta)$ and $\mathbb{E}[\hat{\theta}U]=1$. I am struggling with using ...
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Expected value of ith sample versus expected value of a random variable
Consider a random variable $X$.
When we talk about the expected value of the random variable $X$, we use the notation $\mathbb{E}\left(X\right)$.
However, I found that, in introductory statistics ...
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Why is every predicted $y_i$ in linear regression equal to $E[Y|X]$?
I have some loose intuition with this but I don't understand it
Say we have a scatterplot of data in 2 dimensions. Then we can propose a mean model where, for all $x_i \in X$ the estimated $y_i = E[Y]$...
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