Questions tagged [expectation]

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How to link Wasserstein metric to the expectaion value or Euclidean metric?

I need to find an upper bound for Wasserstein between two distributions, $u$ and $v$. I have a bound for their expectation value ($E(u-v)<\delta$), I was wondering can I say $W_2(u,v)\leq E(u-v)<...
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1 vote
1 answer
93 views

The maximum trace of a covariance can be achieved by a discrete random vector?

Given a random variable $X$, satifying $P(0\leq X \leq 1)=1$, and $\mathsf{E}[X^2] = \alpha$. We know its maximum variance $\text{Var}(X) = \alpha(1-\alpha)$ achived by a binary random variable $P(X =...
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2 votes
1 answer
110 views

Is $\lim_{s \rightarrow 1} E(f(X_s)) = \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{k=1}^N f(k)$?

Let $s>1$ be a real number. We look at the zeta probability function / Zipf probability function defined as: $$P(X = n) = \frac{1}{n^s \zeta(s)}$$ Suppose $f: \mathbb{N} \rightarrow \mathbb{R}$ is ...
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1 vote
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18 views

Сonditional characteristics with respect to a discrete random variable [closed]

160 asymmetrical coins participate in the first roll. In the second roll, only those coins on which the "eagle" fell out in the first roll participate. It is known that the probability of an ...
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1 answer
70 views

Integral form of expectation with respect to complex random variables [closed]

Let $h$ be a random variable and $g(h)$ be a real-valued function of $h$. We know that if h is a real-random variable then: $E_h[g(h)] = \int_{-\infty}^{\infty} f(h) g(h) dh$ where f(h) is the PDF of ...
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1 vote
1 answer
51 views

Expected matrix created from two random orthogonal-projection matrices

Consider an arbitrary finite set of orthogonal-projection matrices (symmetric, idempotent, etc.) in $\mathbb{R}^{n\times n}$. We draw two matrices $Q,P$ uniformly and i.i.d. from this set. Question: ...
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1 vote
1 answer
61 views

Partial derivative of expectation and Stein's lemma

Currently, I am reading a paper about the Gaussian Process in Neural Network [1]. In the solution of the main result in this paper, the author applied Stein's lemma and claimed an equation about the ...
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39 views

Expected value of ceiling of a random variable

I have a continuous non-negative random variable $X \ge 0$ defined by a black-box cumulative distribution function $F(x) = \Pr [ X \le x ]$. In other words, I have an algorithm to calculate $F(x)$ for ...
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65 views

Concentration of number of vertices in convex hull of random points

Dwyer showed that given a set of $n$ points $P$ which were uniformly and independently sampled from the unit cube $[0,1]^d$, if $X$ is the number of vertices in the convex hull of $P$, then $$\mathbb{...
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4 votes
0 answers
219 views

A tricky integral equation

In the context of reconstruction of climate data from ice cores (see related question at MSE) I came about the following problem. (More background and motivation can be given on demand.) Let $f:\...
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3 votes
2 answers
180 views

A lower bound for the expectation of $\min\{X,n-X\}$ when $X$ follows a $\mathrm{Binomial}(n,p)$ distribution

Let $X$ be a random variable following a $\mathrm{Binomial}(n,p)$ distribution, and let $$Y=\min\{X,n-X\}.$$ Ispired by the problem posed by C. Clement on https://math.stackexchange.com/questions/...
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3 votes
1 answer
95 views

Derivative of an integral of a Gaussian

I'd like to compute the derivative of an expected value w.r.t one of the parameters that define the mean of a Gaussian: $ Z=\int \mathcal{N}(x;\mu,\Sigma)f(x) \, dx $, then $ \frac{dZ}{dK}=\text{??}$ ...
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7 votes
1 answer
276 views

Expectation for game choosing uniformly number in $[0,1]$ until it decreases

We are playing a game where we keep on choosing a number from the uniform distribution U(0,1). The game goes on until we have the current number less than the previously picked number, i.e. the game ...
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2 votes
1 answer
79 views

Inequality for increments of $r$th absolute moments of martingales, $1<r<2$

If $Y_n=\sum_{i=1}^n X_i$ is a martingale, where $X_i$ is a martingale difference sequence, $\mathbb{E}[X_n\mid \mathcal{F}_{n-1}]=0$ for all $n$, we know that $$ \mathbb{E}\big[Y_n^2-Y_{n-1}^2\big]=\...
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1 answer
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expectation of log(1-x^a) if x is a beta random variable

How can I compute $\mathbb{E}_{q}\Big[\log (1-x^a)\Big]$ when the distribution of $q$ is given as $q(x)\sim\mathrm{Beta}(\alpha,\beta)$?
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1 answer
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Random walks on GW-trees (transformation)

Let $(X_n)_{n\in\mathbb{N}_0}$ be a biased Random Walk on Galton-Watson tree with $\lambda\in(\lambda_c,m)$. How can I obtain the following equation: $\sum_{k=0}^{n-1}\mathbb{E}_{e_*}[|X_{k+1}|-|X_k| \...
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1 vote
1 answer
249 views

Stochastic Integral + conditional expectation

Let $\overline{\widehat{Z}_i} = \frac{E_i\left[ \int_{t_i}^{t_{i+1}}\widehat{Z}_sds\right] }{\Delta t_i}$ with $\widehat{Z}$ a square integrable process, $\Delta t_i := t_{i+1} - t_i$, and $E_i$ ...
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112 views

What is known about the maximum of several independent standard normal random variables?

Bailey et al., in the May 2014 issue of Notices of the AMS, show that, if $X_k\,$ ($k=1,...,n$) are independent standard normal random variables, then the expectation of their maximum is given ...
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2 votes
1 answer
216 views

Expected value of attempts needed to find a "pair" of cards

We are given an integer $n \geq 1$ and $2n$ cards, labelled $0$ to $2n-1$. We pick a card with uniform probability, put it back, and continue, until for some $k\in \{0,n-1\}$ the cards $2k$ and $2k+1$ ...
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0 votes
0 answers
56 views

EM algorithm for Factor Analysis

I've just learned EM algorithm and Factor Analysis. However, when it comes to applying EM algorithm to Factor analysis, I get confused. It would be much appreciated if someone could help me out. Based ...
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1 vote
0 answers
69 views

Calculating the mean squared error for an estimate of a large sum

Consider the set of all Boolean function $f: \{0, 1\}^{n} \rightarrow \{-1, 1\}$. Now, let's pick a function uniformly at random from this set. Let $F$ be the random variable corresponding to the ...
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0 votes
1 answer
201 views

Product of three or more independent sub-Gaussian varibles

A random variable $X$ is called subgaussian of order $\sigma^2$ if $\log E[exp\{\theta X\}]\leq \frac{1}{2}\theta^2\sigma^2$ for every $\theta\in\mathbb R$. Given a sequence of independent subgaussian ...
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9 votes
1 answer
316 views

Bounds on the expectation of $|X-Y|$ for $X,Y$ Poisson

I would have a proof of the following fact; but it's a bit clunky, and am wondering if one can get a more elegant one (and/or improve the constants). I couldn't find this anywhere, and searching ...
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1 vote
0 answers
201 views

Expected value of length of longest cycle in permutation

Let $n$ be a positive integer and let $S_n$ be the collection of permutations $\pi:\{1,\ldots,n\}\to \{1,\ldots,n\}$. For $\pi\in S_n$ let $\text{maxcyc}(\pi)$ denote the length of the longest cycle ...
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2 votes
0 answers
105 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 ...
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1 vote
0 answers
98 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 ...
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1 vote
0 answers
116 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 ...
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1 vote
1 answer
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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'' $...
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5 votes
2 answers
208 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 ...
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0 votes
1 answer
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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 $\...
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2 votes
1 answer
73 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 ...
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8 votes
3 answers
368 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 ...
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  • 3,668
3 votes
1 answer
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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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2 votes
0 answers
289 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-...
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0 votes
1 answer
74 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 ...
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0 votes
0 answers
84 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 ...
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2 votes
0 answers
36 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 $m$...
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12 votes
4 answers
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, ...
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10 votes
2 answers
209 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 ...
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0 votes
1 answer
221 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 ...
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1 vote
2 answers
119 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 } \...
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0 votes
2 answers
173 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 ...
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1 vote
0 answers
61 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\...
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0 votes
1 answer
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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{...
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2 votes
0 answers
59 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 $...
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2 votes
2 answers
210 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 ...
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1 vote
1 answer
109 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]...
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  • 123
4 votes
1 answer
197 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 ...
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  • 143
-2 votes
1 answer
105 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^...
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-2 votes
1 answer
150 views

If a sequence $X_n$ of RVs converges in probability to $X$, does the sequence $\mathbb{E}(X_n)$ also converge to $\mathbb{E}(X)$? [closed]

I couldn't find the answer in literature so any idea would be helpful.
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