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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 ...
Sumit Singh's user avatar
0 votes
0 answers
33 views

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 ...
user7341333's user avatar
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 ...
Keven Sun's user avatar
2 votes
1 answer
61 views

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_{...
Daniel Cortild's user avatar
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0 answers
31 views

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 ...
kvphxga's user avatar
  • 187
1 vote
0 answers
33 views

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 ...
Grigori's user avatar
  • 33
0 votes
0 answers
237 views

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(...
D. S.'s user avatar
  • 77
1 vote
1 answer
97 views

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 $\mathbb{F}_q$, ...
en-drix's user avatar
  • 125
3 votes
1 answer
244 views

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 $\int_X x dp$ ...
J.R.'s user avatar
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380 views

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: $...
Jim Chen's user avatar
1 vote
0 answers
104 views

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 ...
Dominic van der Zypen's user avatar
3 votes
1 answer
195 views

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 ...
Jean Daviau's user avatar
9 votes
1 answer
696 views

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 ...
fengzju's user avatar
  • 99
14 votes
1 answer
988 views

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 ...
Dan's user avatar
  • 2,997
-1 votes
1 answer
124 views

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 ...
Yauhen Yakimenka's user avatar
8 votes
3 answers
569 views

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 ...
TheSimpliFire's user avatar
0 votes
2 answers
215 views

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 $K \times K$ ...
Dalek's user avatar
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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 \...
user1172131's user avatar
2 votes
0 answers
73 views

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-...
Lucas's user avatar
  • 21
5 votes
1 answer
460 views

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}\...
Aryeh Kontorovich's user avatar
5 votes
2 answers
161 views

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}}\!\...
Itay's user avatar
  • 723
0 votes
1 answer
88 views

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 \...
karel's user avatar
  • 11
0 votes
0 answers
69 views

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 $\...
Prokins Wang's user avatar
0 votes
1 answer
304 views

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 ...
Arbuja's user avatar
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1 vote
0 answers
113 views

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 $A\subseteq X$,...
Arbuja's user avatar
  • 1
6 votes
2 answers
261 views

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, ...
Itay's user avatar
  • 723
1 vote
0 answers
732 views

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 ...
Arbuja's user avatar
  • 1
13 votes
2 answers
368 views

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 ...
Daniel Weber's user avatar
  • 3,064
0 votes
1 answer
81 views

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, $x=a$, ...
dhp's user avatar
  • 11
2 votes
1 answer
186 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,\...
Star's user avatar
  • 76
0 votes
1 answer
148 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 ...
Star's user avatar
  • 76
2 votes
1 answer
195 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
191 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
  • 23
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 $\mathbb{P}(X_1=X_2=\ldots=X_n=0)$...
Penelope Benenati's user avatar
0 votes
1 answer
110 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
42 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
0 answers
57 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 ...
fantacy_crs's user avatar
31 votes
5 answers
2k 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
  • 2,997
5 votes
1 answer
590 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
1 vote
0 answers
149 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
192 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
  • 25
0 votes
0 answers
170 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
24 votes
3 answers
921 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
  • 12.8k
0 votes
0 answers
116 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 ...
Star's user avatar
  • 76
1 vote
1 answer
87 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
  • 1,274
0 votes
1 answer
165 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
233 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
131 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
19 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
232 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