# Questions tagged [expectation]

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### Size of an “average” ϵ-net on the unit sphere

This is a question I originally asked on math.stackexchange, but didn't receive a satisfying answer. Let $\epsilon>0$ and consider constructing a set $S_\epsilon\subseteq S^{d-1}$ of points on the ...
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### Pulling random times out of conditional expectation (“Substitution rule”)

Problem Let $G$ be a positive random variable (a random time) that is a.s. finite, $(X)_{t \geq 0}$ be a càdlàg process taking values in $\mathbb{R}^d$ and $g$ is some sufficiently nice real-valued ...
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### Relation satisfied by a Gaussian random variable

I want to prove the following relation for $X\sim \mathcal{N}(0,1)$, $x\in \mathbb{R}$ and $f(x)=\mathbb{E}[\max(X,x)]$: $$f(\frac{f(x+1)+f(x-1)}{2})\leq \frac{f(f(x)-1)+f(f(x)+1)}{2}$$ It seems that ...
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### Judge a special positive definite matrix in probability

Assume $\mathbf{x}$ is a random vector. The question is to judge whether $$E \{ (\mathbf{xx'})^{-1} \}- E\{(\mathbf{xx'})\}^{-1}$$ is positive definite or not. I have no idea how to do it. Could ...
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### Expected distance between two uniform points in distinct rectangles

Are there any good approximations (especially upper bounds) for the quantity $E(\|X_1-X_2\|$), where each $X_i$ is uniformly distributed in a rectangle $[a_i,b_i]\times[c_i,d_i]$? It does not appear ...
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### Memory game inspired problem

Motivation. As I was playing the pairs-matching game "Memory" (known as "Concentration" in some parts of the world) with my children, I was surprised that even thorough shuffling ...
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### For the following class of matrices, are the determinants invariant under permutations?

I want to ask a question regarding the invariance of determinants under permutation. The following matrix is the one I want to discuss here. (It's just a symmetric block tridiagonal matrix with non-...
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### Expectation of ratios of probability density functions

I'm trying to solve/simplify the expression $$\mathbb{E_{x \sim b(x)}} B\ [\log\left(1 - \frac{A\ a(x)}{2\ c(x)}\right)],$$ or $$B \int_{x}b(x)\log\left(1 - \frac{A\ a(x)}{2\ c(x)}\right)dx,$$ where ...
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### Question about the intensity of a cox process, Diggle–Moraga–Rowlingson–Taylor (2013)

On page 2 of "Spatial and spatio-temporal log-Gaussian Cox processes: Extending the geostatistical paradigm" by Diggle–Moraga–Rowlingson–Taylor (2013), accessible at arXiv, they claim the ...
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### Definition of conditional expectation for singleton

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $\mathcal{G} \subset \mathcal{F}$ be a sub-$\sigma$-algebra. Furthermore, let $X, Y$ be two random variables from our probability ...
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### What are the expected values of the volumes of two classes of ellipsoids contained within the unit 3-ball, and/or what is their ratio?

Consider the class of all ellipsoids contained in the unit 3-ball, and also the subclass of those ellipsoids also contained within tetrahedra also contained in the unit 3-ball. What are the expected ...
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### Expectation of Hadwiger number of a random graph

For any integer $n$, let ${\cal G}_n$ denote the set of simple, undirected graphs $G = (V, E)$ where $V = \{1,\ldots,n\}$. The Hadwiger number $\eta(G)$ of a finite graph $G$ is the maximum integer $m$...
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### Throwing a fair die until most recent roll is smaller than previous one

I roll a fair die with $n>1$ sides until the most recent roll is smaller than the previous one. Let $E_n$ be the expected number of rolls. Do we have $\lim_{n\to\infty} E_n < \infty$? If not, ...
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### The expectation of binary logistics regression with respect to Gaussian distribution

I am trying to compute the expectation of $g(s,x)=s \ln \sigma(x)+(1-s)\ln(1-\sigma(x))$ with respect to the normal distribution $\mathcal{N}(x;m,v)$, where we have $\sigma(x)=\frac{1}{1+e^{-x}}$. If ...
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### Expectation of a linear operator

We define $T: C[0,1]\to C[0,1]\ni T(f(x))= \sum\limits_{k=1}^{m} p_k (f\circ f_k)(x):=\mathbb E( f(X_{n+1}|X_n=x)$ for a system $X_{n+1}=f_{\omega_n}(X_n), n=0,1,2\dots.$ and $\omega_n$ are i.i.d ...
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Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a probability space. Let $$X, Y : \Omega \rightarrow \mathbb{R}$$ be random variables. Furthermore, let $$f: \mathbb{R}^2 \rightarrow \mathbb{R}$$ be a $... 1answer 635 views ### Is there a tight lower bound for the expectation of the product of two positive valued random variables? Let$X,Y$be two (dependent) random variables with$\mathbb{P}(X\ge 0)=\mathbb{P}(Y\ge 0)=1$. I want to find a tight lower bound of$\mathbb{E}(XY)$when$X,Y$are non-negative, almost surely. ... 2answers 182 views ### Expected value of length of interval game I have a die that produces uniformly distributed values in$\{1,\ldots, k\}$for some integer$k\geq 2$. Now I play the following game. I start rolling the die and produce one integer in$\{1,\ldots,...
It is well known that $E[X|X+Y]$ is Gaussian if both $X$ and $Y$ are, and the result can be derived using standard density arguments. However, how can one prove it by only resulting to optimization ...