Questions tagged [expectation]

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2
votes
0answers
67 views

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 ...
1
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0answers
82 views

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 ...
1
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0answers
114 views

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 ...
1
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1answer
122 views

Expectation of maximum of all period lengths of functions $f:\{1,\ldots,n\}\to\{1,\ldots,n\}$

This is based on an older question. For $n\in\mathbb{N}$, let $[n]:= \{1,\ldots,n\}$. Let $\text{Fun}(n)$ denote the set of all functions $f:[n]\to[n]$. To $f\in\text{Fun}(n)$ and ''starting value'' $...
5
votes
2answers
190 views

Expectation of period length of functions $f:\{1,\ldots,n\}\to \{1,\ldots,n\}$

For $n\in\mathbb{N}$, let $[n]:= \{1,\ldots,n\}$. Let $\text{Fun}(n)$ denote the set of all functions $f:[n]\to[n]$. To $f\in\text{Fun}(n)$ associate a sequence $\text{seq}(f))$ defined recursively by ...
0
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1answer
35 views

Simplification on the estimation on error of the ratio of 2 random variables

Let $Z=\dfrac{X}{Y}$ the ratio of 2 random variables. Distribution of $Z=\dfrac{X}{Y}$ Consider the case of two independent normal variables $X$ and $Y$ with strictly positive means and variances $\...
2
votes
1answer
70 views

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 ...
8
votes
3answers
322 views

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 ...
3
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1answer
141 views

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 ...
2
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0answers
264 views

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-...
0
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0answers
34 views

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 ...
0
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1answer
57 views

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 ...
0
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0answers
67 views

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 ...
0
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0answers
42 views

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 ...
2
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0answers
33 views

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 $m$...
12
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4answers
2k views

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, ...
0
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0answers
38 views

Expectation of random matrix (minimum positive eigenvalue)

Assume, $M_i$ are random symmetric positive semi-definite matrices and there exists $ 0 < \mu_1 \leq \mu_2 $ such that $\mu_1 \|x\|^2 \leq x^TE[M_i]x \leq \mu_2 \|x\|^2$ holds for some $0 \neq x \...
10
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2answers
180 views

Maximal in-degree in directed voting graph

Real-life motivation. Our team has $n$ members. For the next in-team presentation session, everyone had 1 talk prepared that he or she would be able to present. Now everyone could cast $1$ vote about ...
0
votes
1answer
144 views

Concentration of $\ell_2$ norm of a vector sampled from a distribution

Let $X=(X_1,\ldots,X_n)$, where $X_i \sim P_{p_i}(0,\frac{1}{\lambda})$ are iid, $P_{p_i}$ is sub gaussian distribution for $i^\text{th}$ element, and 0 and $1/\lambda$ are mean and variance. I'm ...
1
vote
2answers
108 views

Inaccurate results for the analytical expression of $\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right]$

I'm trying to plot a graph for the following expectation $$\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right]=a 2^{-\frac{\kappa }{2}-1} b^{-\frac{\kappa }{2}} \theta ^{-\kappa } \...
0
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2answers
100 views

Finding the expectation of $a \mathcal{Q} \left( \sqrt{b } \gamma \right) $, where $\gamma$ is a Gamma r.v

I'm trying to analytically find the following expectation $$\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right],$$ where $a$ and $b$ are constant values, $\mathcal{Q}$ is the ...
1
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0answers
44 views

Negative moments of Steinhaus random variables

Let $f_i, i=1, \ldots, n$ be independent Steinhaus random variables, i.e. variables which are uniformly distributed on the complex unit circle. Let $a \in R^n$. 1) Find $E\left(\sum_{i=1}^nf_i a_i\...
0
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1answer
77 views

expectation of the function of Wishart matrix eigenvalues

For Given a $N×M$ random complex gaussian matrix $X$ where $M=XX^H$, let $\lambda_1>\lambda_2>....>\lambda_N$ be the ordered eigenvalues of $M$ my objective is to get an estimation of \begin{...
2
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0answers
49 views

The expected size of a subtree of any labelled rooted tree

Consider the set of labelled rooted trees of size $n$, $\mathcal{T}_n$. Let $r$ be the root of each tree $T=(V,E)\in\mathcal{T}$, $r\in V(T)$, and let $n(u)$ be the number of vertices of the subtree $...
2
votes
2answers
131 views

Disintegration, conditional probabilities, and conditional expectation

On the Wikipedia page there is a note that conditional probability measures can be described by disintegration. However, I can seem to find a clear exposée of how this construction is related to ...
1
vote
1answer
60 views

An attempt to find expected value of clique number of special random graph

Let $G(n)=(V,\mathcal{E})$ be a random graph definded as follows: $V=[n]=\{1,2, ... ,n\}$ and for all $i,j\in V$ so that $i\ne j$ we have $\{i,j\}\in\mathcal{E}$ with probability $p$. Where $p\in[0,1]...
4
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1answer
186 views

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 ...
-2
votes
1answer
104 views

In which cases $E(e^{t S_n S_m})$ converges to $E(e^{t X Y})$

Suppose that $S_n$ and $S_m$ are two random binomial variables, which are independent and with the same distribution parameter $p$. I am wondering, in which cases $E(e^{t S_n S_m})$ converges to $E(e^...
-2
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1answer
75 views
2
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0answers
49 views

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 ...
1
vote
1answer
61 views

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 ...
4
votes
2answers
475 views

Bounding an expectation involving i.i.d. standard Gaussians and Rademacher

I have tried to bound the following quantity, but cannot get the "right" (conjectured) bound: $$ \phi(\gamma,d,n) = -1+e^{\frac{1}{2}n\gamma^2 d} \mathbb{E}_{X}\left[\frac{\mathbb{E}_Z[\prod_{j=1}^n(...
0
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0answers
38 views

moment generation function for matrix Gaussian distribution

What is the moment generation function for the following function $$ E(e^{\mathbf{X}^T\mathbf{y}\mathbf{W}})=\int e^{\mathbf{X}^T\mathbf{y}\mathbf{W}}\frac{\exp\big(-\frac{1}{2}\mathrm{tr}\big[\mathbf{...
0
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1answer
132 views

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 ...
0
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1answer
42 views

expectation of a quadratic function of a matrix variate normal distribution

I want to compute the following expectation term: $E[{\bf{XA}}{{\bf{X}}^T}]$ where ${\bf X} \in R^{M \times M}$ and its elements are normal random variables such that $vec\left( {\bf{X}} \right)\...
0
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1answer
117 views

Are there known bounds on the expectation of the truncated Beta distribution?

Let $X\sim beta(\alpha,\beta)$ be a random variable and let $\tau\in(0,1)$. Are there any known closed-form bounds (I'm specifically interested in lower bounds) on $$ \mathbb E[X\ | X\le \tau]? $$
1
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1answer
135 views

upper bound of the expectation of upper quantile ratio

We have a collection $\boldsymbol{S}$ of $n$ discrete random variables $X_1$, $X_2$, $\dots$, $X_n$ $\overset{\small \text{i.i.d.}}{\small \sim}$ $\mathcal{D}$, where $\mathcal{D}$ is a distribution ...
-1
votes
1answer
175 views

expectation of upper quantile proportion

(edited considerably following comments) We have a collection $\boldsymbol{S}$ of $n$ discrete random variables $X_1$, $X_2$, $\dots$, $X_n$ $\overset{\small \text{i.i.d.}}{\small \sim}$ $\mathcal{D}$...
3
votes
1answer
151 views

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 ...
1
vote
2answers
68 views

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 $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ ...
7
votes
1answer
391 views

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?
0
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0answers
71 views

A closed form of mean-field equations

Assume that a system at time t, for example number of costumers in a line at time $t$ which is denoted by $q(t)$, follows a Markov chain with these dynamics (probabilities) $$P(q(t+\Delta t)-q(t)=1)=\...
0
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0answers
68 views

Root of the expectation of a random rational function

I am trying to figure out a formula for the unique $\lambda>1$ such that $$ \mathbb{E}\bigg[\frac{X}{\lambda -X}\bigg]=1 $$ where $X$ is a discrete random variable taking values in $\{\frac{1}{n},....
1
vote
1answer
92 views

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 ...
1
vote
1answer
239 views

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 ...
2
votes
2answers
173 views

Substitute Concrete Value in Conditional Expectation

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 $...
2
votes
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. ...
3
votes
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,...
0
votes
1answer
68 views

CDF of a RV that is the ratio between a complex Gaussian and a Chi-squared RVs

Given the following p.d.f., which is the p.d.f. of the real and imaginary parts of a random variable that is the ratio between a complex Gaussian and a Chi-squared RVs: \begin{equation*} f_U(u)=\exp\...
1
vote
1answer
85 views

Minimization Proof of Conditioning on Gaussian is Gaussian

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 ...