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Questions tagged [expectation]

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Methods for high-dimensional integration of joint PDFs over unbounded regions

I am currently grappling with an issue related to high-dimensional integration of joint Probability Density Functions (PDFs) over unbounded regions. The specifics of the problem are a bit intricate, ...
scmath's user avatar
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2 votes
1 answer
147 views

Bound the probability that a point belongs to a set

Let $(a_k)_{k \geq 1}$ be random variables taking values on a finite subset $B$. Assume that $$ (1) \quad \Pr\Big (\lim_{n\rightarrow +\infty}d(\frac{1}{n}\sum_{k=1}^n 1_{[a_k = b]}, [v_\ell(b,\...
TEX's user avatar
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0 votes
1 answer
63 views

Bound the expectation of an average

Let $(a_n)_{n \geq 1}$ be random variables taking values on a finite subset $B$. Assume that $\nu_l(b) \le P[a_n = b\mid a_1,\ldots,a_{n-1}] \le \nu_u(b)$ almost surely for every $n \ge 1$ and $b \in ...
TEX's user avatar
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2 votes
1 answer
186 views

Tossing a coin around $\mathbb{Z}/n\mathbb{Z}$ [closed]

Motivation. With my younger son I played the following game on a big (dysfunctional) clock which can be modelled as $\mathbb{Z}/12\mathbb{Z}$ : Put the clock hands at number $12 ( = 0)$. Toss a coin, ...
Dominic van der Zypen's user avatar
1 vote
1 answer
113 views

Expected (maximum minus minimum) of Laplacian random variables

Suppose there are $n$ IID random variables denoted as $X=(X_1,\dots, X_n)$, they follow Laplace distribution with parameter $\lambda$, denoted as $Lap(\lambda)$. That is, $$f(x)=\frac{1}{2\lambda}\exp ...
white's user avatar
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2 votes
1 answer
263 views

On the mean value taken by Bernoulli random variables with joint distribution constraints

We are given a vector $n$-dimensional random vector $\mathbf{X}$ whose components are the Bernoulli random variables $X_1, X_2, \ldots X_n$, such that the probability $\mathbb{P}(X_1=X_2=\ldots=X_n=0)$...
Penelope Benenati's user avatar
0 votes
1 answer
74 views

Approximation for an expectation expression

Let $\mathbf{x} \in \mathbb{C}^M$ is an unknown distributed random vector (certainly not gaussian), and matrix $\mathbf{A}\in \mathbb{C}^{M \times M}$ which is fix (known). Also, assume we know the ...
A. R.'s user avatar
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2 votes
0 answers
35 views

can we get a family of classifiers $\left\{f_n\right\}_{n \in N}$such that $\lim_{n->∞} (E_{(X_1, Y_1), ...,(X_n, Y_n) \sim \rho}[R(f_n)]-R(f_B))=0 $

For a given classifier $f: \mathbb{R}^d \mapsto\{0,1,2\}$, let $$ R(f):=\mathbb{E}_{(X, Y) \sim \rho}\left[\mathbb{1}_{f(X) \neq Y}\right] $$ $f_B$ the Bayes classifier. can we get a family of ...
fantacy_crs's user avatar
3 votes
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How to prove emprical risk converges to expectation risk as $n\to \infty$?

For example, for a classical binary classification: $x \in \mathbb{R}^d$ and $y \in\{0,1\}$ let empirical risk be $R_{\ell}^n(f):=\frac{1}{n} \sum_{i=1}^n \ell\left(f\left(X_i\right), Y_i\right)$ and ...
fantacy_crs's user avatar
31 votes
5 answers
1k views

On average, how many uniformly random real numbers $u$ are needed for their sum to exceed $1$, if $u_1$ is in $(0,1)$ and $u_k$ is in $(0,eu_{k-1})$?

A well-known question is: on average, how many uniformly random real numbers in $(0,1)$ are needed for their sum to exceed $1$? The answer is $e$. Let's tweak this question by making each random ...
Dan's user avatar
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5 votes
1 answer
190 views

Maximum distance from origin of simple random walk

Let $\epsilon_1, \dots, \epsilon_n$ be random signs, equiprobably in $\{-1, 1\}$, independently. Let $S_k = \sum_{j=1}^k \epsilon_j$. I am wondering what is known about the expectation $$ \mathbb{E}\...
Drew Brady's user avatar
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0 answers
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Taylor series for multidimensional orthant probability

Here we have a solution as Measuring Solid Angles Beyond Dimension Three, https://doi.org/10.1007%2Fs00454-006-1253-4. As we know there is no closed-form expression for the probability of higher than ...
A. R.'s user avatar
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1 vote
0 answers
95 views

Expectation of inverse of complex Gaussian variables

If we consider a complex Gaussian random variable as $h\sim\mathcal{CN}(0,\gamma)$, where $\gamma$ is the variance. Is there any closed-form solution with $\gamma$ for $\mathbf{E}\left[\frac{1}{\lVert ...
Charlie Nie's user avatar
1 vote
1 answer
135 views

Probability of multivariant gaussian random variables in different areas

$\newcommand{\sgn}{\operatorname{sgn}}$Let $X_i$ is a gaussian random variable correlated with others. we want to find the probability of each possible case to find the expectation of following ...
A. R.'s user avatar
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0 votes
0 answers
47 views

Is $\mathbb E [zz^T / z^Tz] = I / n$ is generally well known equation when $z \sim N(0,I)$, $I \in \mathbb{R}^{n \times n}$?

I found that following equation holds for random vector $z \sim N(0,I)$ : $\mathbb{E} [\frac{zz^T}{z^Tz}] = \frac{1}{n} I$ Proof is very simple that is only calculating integral for each component ...
GrossMatt's user avatar
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0 answers
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Expectation of a norm in Monte-Carlo method

I am currently studying the Monte Carlo methods for solving PDEs with random coefficients. My problem here is basically just doing with some algebraic properties of the expected value function which I ...
user492649's user avatar
0 votes
0 answers
156 views

How many elements have a "small" order in a finite field?

I'm hoping that this is an easy question for someone. How many elements can we expect to have multiplicative order at most $n^{1/c}$ in one of the finite fields $\mathbb{F}_p$ with $p$ prime with $n \...
Matt Groff's user avatar
23 votes
3 answers
890 views

Are there arbitrarily large families of lines in $\Bbb R^3$ with average angle $\ge \pi/3$?

Question: Can I have an arbitrarily large finite family of lines $\ell_1,\dotsc,\ell_n\subset\Bbb R^3$ so that the average angle between two (distinct) lines is $\ge \pi/3$? We can assume that all ...
M. Winter's user avatar
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0 votes
0 answers
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Does $E(XU)\neq 0$ imply $E(f(X) U)\neq 0$ "almost always"?

Consider two non-orthogonal random variables $$ (1) \quad E(XU)\neq 0, $$ where $X$ can be a vector. Can we claim that (1) implies that $U$ will be "generically" non-orthogonal to any ...
TEX's user avatar
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0 answers
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Computing the expectation of $\mathrm{tr}((W + A)^{-1})$ where $W$ is a random Wishart matrix?

Let $W \sim \mathcal{W}_d(V, n)$ be a random Wishart matrix. Let $A$ be a real symmetric positive definite matrix. I am interested in computing $$ \varphi_{V, d, n}(A) := \mathrm{tr}\big[\mathbb{E}[(W ...
Drew Brady's user avatar
1 vote
1 answer
75 views

Is this expectation $\mathbb E\big[{\bf 1}_{\{x+\inf_{0\le t\le 2}W_t>0\}}(W_{\tau}-y)^+\big]$ strictly positive?

Let $(W_t)_{t\ge 0}$ be a standard Brownian motion and $\tau$ be a stopping time lying in $[1,2]$. For $x, y>0$, can we show $$\mathbb E\big[{\bf 1}_{\{x+\inf_{0\le t\le 2}W_t>0\}}(W_{\tau}-y)^+\...
GJC20's user avatar
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0 votes
1 answer
122 views

Union Bound of two events?

Inequality 1 \begin{align} \mathbb{P}\left(\frac{1}{n} \sum_{i=1}^{n}\left(f\left(x_{i}\right)-\mathbb{E}[f]\right) z_{i} \geq \frac{\epsilon}{8}\right) \leq 2 \exp \left(-\frac{\epsilon^{2} n d}{9^{4}...
user avatar
1 vote
1 answer
126 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 =...
Jone Sweden's user avatar
2 votes
1 answer
122 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 ...
mathoverflowUser's user avatar
1 vote
0 answers
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 ...
Ben's user avatar
  • 11
0 votes
1 answer
143 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 ...
Bertrille's user avatar
2 votes
1 answer
95 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: ...
Itay's user avatar
  • 547
1 vote
1 answer
167 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 ...
Quicky2357's user avatar
0 votes
0 answers
101 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 ...
Yauhen Yakimenka's user avatar
4 votes
0 answers
234 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:\...
Hans-Peter Stricker's user avatar
3 votes
2 answers
311 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/...
Xueyi Huang's user avatar
3 votes
1 answer
163 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{??}$ ...
GuyS's user avatar
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7 votes
1 answer
298 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 ...
Shashank Nathani's user avatar
2 votes
1 answer
162 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]=\...
mattia's user avatar
  • 23
0 votes
1 answer
260 views

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 $q(x)\sim\mathrm{Beta}(\alpha,\beta)$?
Dalek's user avatar
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0 votes
1 answer
62 views

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| \...
Fynn13's user avatar
  • 65
1 vote
1 answer
421 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 $E_i$ ...
freshst4r's user avatar
1 vote
0 answers
721 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 ...
John Bentin's user avatar
  • 2,417
2 votes
1 answer
237 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 $2k+1$ ...
Dominic van der Zypen's user avatar
1 vote
0 answers
75 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 ...
RandomMatrices's user avatar
0 votes
1 answer
459 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 ...
user80904's user avatar
9 votes
1 answer
360 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 ...
Clement C.'s user avatar
  • 1,282
2 votes
0 answers
419 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 ...
Dominic van der Zypen's user avatar
2 votes
0 answers
129 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 ...
R B's user avatar
  • 608
1 vote
0 answers
119 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 ...
Probability Boi's user avatar
1 vote
0 answers
118 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 ...
Pierre's user avatar
  • 171
1 vote
1 answer
136 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'' $...
Dominic van der Zypen's user avatar
5 votes
2 answers
245 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 ...
Dominic van der Zypen's user avatar
0 votes
1 answer
160 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 $\...
youpilat13's user avatar
2 votes
1 answer
79 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 ...
Regan's user avatar
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