# Questions tagged [expectation]

The expectation tag has no usage guidance.

112
questions

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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, ...

2
votes

1
answer

147
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### 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,\...

0
votes

1
answer

63
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### 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 ...

2
votes

1
answer

186
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### 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, ...

1
vote

1
answer

113
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### 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 ...

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 $\mathbb{P}(X_1=X_2=\ldots=X_n=0)$...

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 ...

2
votes

0
answers

35
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### 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 ...

3
votes

0
answers

45
views

### 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 ...

31
votes

5
answers

1k
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### 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 ...

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}\...

0
votes

0
answers

36
views

### 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 ...

1
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0
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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 ...

1
vote

1
answer

135
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### 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 ...

0
votes

0
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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 ...

0
votes

0
answers

51
views

### 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 ...

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 \...

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 ...

0
votes

0
answers

63
views

### 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 ...

0
votes

0
answers

30
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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 ...

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)^+\...

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}...

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 =...

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votes

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answer

122
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### 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 ...

1
vote

0
answers

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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 ...

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 ...

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: ...

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 ...

0
votes

0
answers

101
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### 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 ...

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:\...

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/...

3
votes

1
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163
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### 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{??}$ ...

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 ...

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]=\...

0
votes

1
answer

260
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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 $q(x)\sim\mathrm{Beta}(\alpha,\beta)$?

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| \...

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 $E_i$ ...

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 ...

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 $2k+1$ ...

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 ...

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 ...

9
votes

1
answer

360
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### 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 ...

2
votes

0
answers

419
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### 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 ...

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 ...

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 ...

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 ...

1
vote

1
answer

136
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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'' $...

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
...

0
votes

1
answer

160
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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 $\...

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 ...