# Questions tagged [expectation]

The expectation tag has no usage guidance.

136
questions

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### Find the Cov$(Z_n^i,Z_n^j)$ where $Z_n^i=\min\{N_i,p\}$, where $N_i$ is binomial

Suppose that $n$ particles (a fixed quantity) are randomly distributed in the interval $[0,1]$ and the particles are independently coming from the uniform distribution in $[0,1]$. Now, consider any ...

0
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0
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33
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### How to calculate the expected distance between two points [closed]

I am trying to calculate the expected distance a mobile user. so if $x_{start}$ and $y_{start}$ are the initial positions and $x_{end}$ and $y_{end}$ are the final positions of the mobile user (which ...

0
votes

1
answer

27
views

### Expectation of spectral norm of a diagonal stochastic matrix

There is a diagonal matrix $G(t)\in R^{M\times M}$ where the diagonal elements are independent Bernoulli stochastic variables, satisfying $\mathbb{E}(g_i(t) = 1) = b$, and $ \mathbb{E}(g_i(t) = 0) = 1 ...

2
votes

1
answer

61
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### Connection between Wassertein-2 metric and difference in variance

Given two probability densities $\mu\in\mathcal P(\mathbb R^d)$ and $\nu\in\mathcal P(\mathbb R^d)$, we define their Wasserstein-$p$ metric as
$$
W_p^p(\mu, \nu)=\inf_{\gamma\in \Gamma(\mu, \nu)}\int_{...

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31
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### Expected (log)volume of projection of a cube onto a random subspace

Suppose $A$ is a full-rank $d\times d$ dimensional matrix. Let $U \in R^{d\times k}$ be a projection to projection matrix onto a uniformly chosen sub-space of dimension $k$ (for example, they can be ...

1
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0
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33
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### Uniform distribution as argument for copula likelihood

I am reading a well-known paper about copulas by Chen and Fan (2006). Specifically, Proposition 4.2 (see attached), in which all the arguments are uniform $U_{t-1}, U_t$. However, when the copula is ...

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237
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### Uniform distribution on strings

Let $x$ be any binary string $\in (0,1)^*.$
The majority language is given by:
$$\text{MAJ}:=\{x\in (0,1)^*:\sum_{i=1}^ {|x|}x_i>\frac{|x|}{2}\},\text{where $x_i$ is the $i$-th position value(...

1
vote

1
answer

97
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### Expected number of solutions of a random quadratic polynomial system over a finite field

Let $\mathbb{F}_q$ be a field of $q$ elements. Let $a_{i,j,k}$, $b_{i,j}$, $c_i$ ($1 \leq i \leq m$, $1 \leq j \leq k \leq n$) be independent uniformly distributed random variables in $\mathbb{F}_q$, ...

3
votes

1
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244
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### Expectation on a Polish space

I was wondering, if given a Polish space $X$, and given some probability measure $p$ on $X$, can the expectation of an $X$-valued function be taken? In particular, would the integral
$\int_X x dp$ ...

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votes

1
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380
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### Minimum of exponential distribution

Consider $n$ independent random variables $𝑋_𝑖\sim\exp(𝜆_𝑖)$ for $I=1\ldots,n$. Let $\lambda = \sum_{i=1}^n\lambda_i$. Of course, the minimum of these exponential distributions has distribution:
$...

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104
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### Expected value of maximal cycle length in fixed-point free bijections

$\newcommand{\n}{\{1,\ldots,n\}}$
$\newcommand{\FF}{\text{FF}}$
$\newcommand{\lc}{\text{lc}}$
Motivation. A group of my son's peers decided to have a few days of Secret Santa before last year's ...

3
votes

1
answer

195
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### Statistically stationary properties of expectations conditioned on the value of an Ornstein–Uhlenbeck process

Consider the modified Ornstein–Uhlenbeck process
$$\mathop{dx_t}=\theta(y_t-x_t)\, dt+{}\sigma\,dW_t$$
for a standard Brownian motion $W_t$ and $\theta,\sigma\in\mathbb{R}_{>0}$. Let's define the ...

9
votes

1
answer

696
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### The expected value of product of random variables which have the same distribution but are not independent

Given a positive integer $k$, is there a positive real number $c(k)$ such that $\mathbb{E}\left(\prod_{i=1}^k X_i\right)\geq c(k)$ for any $k$-random variables $X_1,X_2,\ldots,X_k$ which all have the ...

14
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1
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988
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### A disc contains many random points. Each point is connected to its nearest neighbor. What is the expectation of average cluster size?

A disc contains $n$ independent uniformly distributed points. Each point is connected by a line segment to its nearest neighbor, forming clusters of connected points.
For example, here are $20$ random ...

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votes

1
answer

124
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### Weighted sum of zero-mean random variables

Let us say we have two independent random variables $X$ and $Y$, with both $E[X] = E[Y] = 0$.
Is it true that for any random weight variable $0 \le W \le 1$ (e.g., $W$ dependent on $X$ and $Y$) we ...

8
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3
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569
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### Jensen-like inequality for random matrix: $\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$

Let $X\in M_n(\Bbb R)$ be a random matrix with iid elements following a continuous distribution.
What are the necessary and sufficient conditions for $$\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$$ to hold? Is ...

0
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2
answers

215
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### Computing the expectation of a quadratic matrix form involving Bernoulli and Gaussian distributed matrices

I am working with two random matrices, $Z$ and $H$:
$Z$ is an $n \times K$ matrix with entries sampled i.i.d. from a Bernoulli distribution: $Z_{ij} \sim \mathrm{Bernoulli}(p)$.
$H$ is a $K \times K$ ...

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44
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### Prove lower collinearity on the tails of Gaussian blob

Let us consider a $n$-dimensional Gaussian blob, i.e. a set of $N$ random vectors $\{\boldsymbol{X}^{(j)}\}_{j=1}^N$, with $n$ independent components, $X_i^{(j)}$, and such that $X_i^{(j)} \sim \...

2
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73
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### Expected number of steps until a queue of $n$ people has passed all $n$ ordered tests consecutively

We are given a queue of $n$ people $\{p_1, \ldots, p_n\}$. They each have to pass $n$ exams $\{t_1, \ldots, t_n\}$. For simplicity we can "draw" the setting in the following way:
$$[t_n,t_{n-...

5
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1
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460
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### Bounding expected maximum via adjacent differences

For $X_i$, $i\in[n]$
be a sequence of integrable random variables.
Is there a universal constant $c>0$ such that
$$\mathbb{E}\max_{i\in[n]}X_i
\le
c\left(
\max_{i\in[n]}\mathbb{E}|X_i|
+
\mathbb{E}\...

5
votes

2
answers

161
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### Expectation of a function of two entries of an isotropic unit vector $\mathbb{E}_{\mathbf{w}\sim\mathcal{S}^{p-1}}\![{w_{1}}^{\!4}\,{w_{2}}^{\!4}]$

Problem:
Given a random isotropic unit vector in $\mathbb{R}^p$ for $p\ge2$, we are trying to compute (preferably exactly, otherwise to upper bound):
$$\mathbb{E}_{\mathbf{w}\sim\mathcal{S}^{p-1}}\!\...

0
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1
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88
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### Inequality on conditional variance of a vector

I have a random vector $X$ and an event $\mathcal{E}$ such that $\mathbb{P}(\mathcal{E}) = p$. I am trying to show the following inequality :
\begin{equation}
p\mathbb{E}[\|X - \mathbb E [X \vert \...

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0
answers

69
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### Expectation of the trace of random matrix with an inverse insided

Consider a $N$-dimensional random complex vector $\mathbf{x} \in \mathbb C^{N \times 1}$ following the complex Gaussian distribution, i.e., $\mathbf{x} \sim {CN}(0,\sigma^2 \mathbf{I})$, where $\...

0
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1
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304
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### Finding examples of functions which are infinite or undefined with current extensions of the expected value?

Preliminaries
Consider the expectations desribed in this paper, which is an extension of the Lebesgue density theorem; this paper which is an extension of the Hausdorff measure, using Hyperbolic ...

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113
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### Are the extensions of the expected value, below, finite for all functions in only a shy subset of all measurable functions?

This is a follow up to this post, where I wish to verify whether one of the statements (in the post) is true but first let's recap the definitions:
Let $(X,d)$ be a metric space. If set $A\subseteq X$,...

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2
answers

261
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### Expectation of the inner product of a subset of two random orthonormal vectors

Setting: Consider sampling two orthonormal vectors $\mathbf{u},\mathbf{v}$ in $\mathbb{R}^p$ (where $p\ge2$) from a "uniform" distribution over the $p$-dimensional sphere (alternatively, ...

1
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0
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732
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### Finding a unique and finite expected value for almost all measurable functions?

Let $(X,d)$ be a metric space. If set $A\subseteq X$, let $H^{\alpha}$ be the $\alpha$-dimensional Hausdorff measure on $A$, where $\alpha\in[0,+\infty)$ and $\text{dim}_{\text{H}}(A)$ is the ...

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2
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368
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### Expected sorting time of random permutation using random comparators

In sorting networks, a comparator of positions $i < j$ is an operator which takes a permutation, checks if $p_i > p_j$, and if it is the case, swaps $p_i$ and $p_j$.
Using this, we can define ...

0
votes

1
answer

81
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### Convergence in expectation of a discontinuous function

Consider a random variable $X\in \mathbb{R}^d$. Let ${\theta_m}$ be a sequence of real numbers that converge to $\theta$. Let $f(x,y)$ be a function that is not continuous. To be specific, fix, $x=a$, ...

2
votes

1
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186
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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
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1
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148
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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

195
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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
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1
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191
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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

318
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
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1
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### 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
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0
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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

57
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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 ...

31
votes

5
answers

2k
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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

590
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### 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}\...

1
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0
answers

149
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### 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

192
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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
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0
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170
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### 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 \...

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3
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921
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### 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
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0
answers

116
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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 ...

1
vote

1
answer

87
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### 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

165
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### 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

233
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### 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 =...

2
votes

1
answer

131
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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
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0
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### 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

232
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### 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 ...