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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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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