# 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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### 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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### 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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