# Questions tagged [expectation]

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### Expectation of multiplied random variables given their individual expectations

Suppose that I have two non-negative real valued random variables $x, y \in Z_+$ that always satisfy $$x+y \leq 1.$$ Also suppose that $E[x] = 1/2$ and $E[y] = 1/4$. What is the maximum possible value ...
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### Approaches and approximations for specific non-convex low-rank optimization problem

I have recently begun looking at an optimization problem in the context of Wiener filtering and lower-dimensional representations. Assume $X$ is a random vector of length $n$ and $Y$ is a random ...
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### An upper bound on $\mathbb{E}\bigg[\bigg(\sum_{i=1}^{k}(X^{\top}A_{i}X)^{2}\bigg)^{q}\bigg]$

Let $X\in\mathbb{R}^{d}$ have independent, mean zero subgaussian entries, and $A_{1},\ldots,A_{k}$ be fixed $d\times d$ matrices that have zeros on the diagonal. I would like to upper bound the ...
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### Conditional density for random effects prediction in GLMM

I am currently working on generalized linear mixed models (GLMM) and need some help concerning the prediction of the random effects. More specifically, I don't understand the given representation of ...
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### Bounds on expectation of X/(X^2 + c) with X ~ Gaussian and c > 0

I'm trying to compute expectation of $X / (X^2 + c)$ when $X$ is normally distributed with mean $\mu$ and variance $\sigma^2$, and $c$ is some positive constant. I think this cannot be solved ...
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### Moment of a function of Gaussian random variables: $\mathbb{E}[(a_{i}^{\top}AA^{\top}a_{j})^{q}]$

Let $A$ be an $m\times k$ matrix with iid $\mathcal{N}(0,1)$ entries and $a_{i}$ and $a_{j}$ be its $i$th and $j$th columns, respectively. I would like to compute the following quantity: \begin{...
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### Expected value of “longest bit / shortest bit” in $n$ uniformly distributed points on $[0,1]$

Let $n\geq 2$ be an integer. We pick $n$ points in $[0,1]$ with uniform distribution. Let $A$ be the minimum distance that two adjacent points have, and let $B$ be the maximum distance that two ...
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### Expectation of $\left| \frac{(\textbf{x}+\textbf{y})^{H} \textbf{x} }{\| \textbf{x} + \textbf{y} \|^2} \right|^2$, with complex Gaussians?

Given that following two random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{x}^{2}\textbf{I}_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ ...
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### Expected value of biggest distance of adjacent points uniformly picked in $[0,1]$

We pick $n\ge 2$ points in $[0,1]$ with uniform distribution. What is the expected value of the largest distance of $2$ adjacent points?
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### Expected value of inverse of complex non-central Wishart matrix

I have a matrix $W$ that abides a complex non-central Wishart distribution. My question is what the expectation of the inverse is, i.e., how to compute $$\mathbb{E}(W^{-1}).$$ I have tried to read up ...
Lets say we get a list $M$ containing $|M|=\sqrt{L\cdot N}$ randomly and independtly drawn elements from a set of size $N$. And lets denote the $i$-th element of the list $M$ by $M[i]$. If we now ask ...
When we handle with some dynamic input-output mappings, there occurs a question as follows: Let $M$ be a random matrix, of which each element contains random terms. Consider the two expectation ...