Questions tagged [probability-distributions]

In probability and statistics, a probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference.

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Why should we expect this odd behavior of negative binomial distributions?

In independent Bernoulli trials with probability $p$ of success on each trial, let $X$ be the number of failures before the $n$th success. Then $$ \Pr(X=x) = \binom{-n}{\phantom{+}x} (-q)^x p^n \text{ ...
Michael Hardy's user avatar
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Kolmogorov inequality for Bernoulli random variables

This question is also asked on math stackexchange. The question is about one inequality which shows in Kolmogorov's paper (inequality (3.1)) but is not proved. The inequality says that, if we assume $...
Greenhand's user avatar
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Understanding simple point processes (part 2)

This is a follow up of this previous question. I'm trying to understand the following proposition from An Introduction to the Theory of Point Processes Volume I: Elementary Theory and Methods by Daley ...
matteogost's user avatar
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Distribution of joint Gaussian conditional on their sum of squares

Given a random gaussian matrix $\mathbf{X}$ with zero mean matrix and covariance matrix $\mathbf{\Sigma}$, and two deterministic matrices $\mathbf{A}$ and $\mathbf{B}$. If I know the value of $\|\...
Sheperd Lv's user avatar
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Understanding simple point processes

Background I'm studying the basic theory of Random Finite Sets (RFS), which is the name that is used in my field to denote simple point processes. A simple point process is a random variable whose ...
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Sum of Skellam-distributed number of random variables

Suppose $X_i$ are i.i.d, and $N \sim \text{Skellam}(\mu_1$, $\mu_2$). Is it possible to find a closed form for the p.d.f of $S_N$, defined by $S_N = X_1 + \cdots X_N$ when $N \ge 0$, and $S_{-N} = -...
Harry L's user avatar
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Some identities from graph theory and probability

The other day I attended a seminar about probability. I took some notes and I am now revising it and trying to understand some steps that were omitted by the lecturer. To formulate my question, ...
MathMath's user avatar
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Concentration inequalities for random sampling without replacement

Let a population $C$ consist of $N$ values $c_1, c_2, \cdots, c_N$, with $c_i\in \{0,1\}$. Let $X_1, X_2, \cdots, X_n$ denote a random sample without replacement from $C$ and let $Y_1, Y_2, \cdots, ...
Dotman's user avatar
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Limit of distributions

Suppose that $X_1,X_2,\ldots, X_n$ are i.i.d random variables with continuous density $f(x)$, which is defined in the whole $\mathbb{R}$. Consider $$s(x)=\lim_{n\to\infty}\frac{1}{n}\log\mathbb{P}(\...
STrick's user avatar
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What's the lower bound of the correlation coefficient?

Suppose a random variable $X \in \mathbb{R}$ follows a discrete distribution $p$ and takes $n$ values. We assume $E[X]=0$ and $|X|\le M$, where $M$ is a constant. Given a smooth and monotonic ...
Jiacai Liu's user avatar
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conjecture for general form of minimax estimator

I had previously posed an overly ambitious version of this conjecture here, Form of minimax estimator, which was quickly shot down by Václav Voráček (on twitter) and Iosif Pinelis (MO answer in the ...
Aryeh Kontorovich's user avatar
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Probability related to record index that cross zeros in $[0, 1, \cdots, N]^{\mathbb{N}}$

This question is inspired by this paper. For those who are interested in more details and applications about record index and record values, you can find them in the paper. Let $X=\{0, 1, 2, \cdots, N\...
Sanae Kochiya's user avatar
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Distribution of waiting time conditioned on a fixed time length

FYI, this question is a duplicate from math stack exchange I ask here again because I got no response. Suppose, I work in a factory production line. The time for me to finish wrapping product $A$ (or $...
Fellow InstituteOfMathophile's user avatar
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Small deviations of real log-concave random variable

I am working with a log-concave real random variable, that has a density $f(x) = \exp(-\varphi(x))$ with $\varphi$ convex. Assuming that $X$ is centered and has unit variance ($\mathbb{E}X=0$, $\...
Hugo Ch's user avatar
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Form of minimax estimator

Let $\Delta$ be the set of all probability distributions over $\mathbb{N}=\{1,2,\ldots\}$ and fix some $\mathcal{P}\subseteq\Delta$. Suppose additionally that $\Delta$ is endowed with some norm $||\...
Aryeh Kontorovich's user avatar
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Distribution of peaks in Dyck paths

A Dyck word is a sequence of open and closed brackets such that the brackets come in correctly matched pairs. For example $(()(()))()$ is a Dyck word, while $())(()$ is not. A Dyck path is a visual ...
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2 votes
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Weak convergence of measures on continuous function spaces

Let $S$ be the unit sphere of $C[0,1], \|\cdot\|_{\infty})$, let $(B_{t})_{t}$ the brownian motion. I would like to show that the measure $\mu_r$ defined on $\mathbb{B}(S)$ by $\mu_r(A):=P\Big(\frac{...
Paul's user avatar
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1 answer
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Extreme confusion with the exact meaning of Gaussian measure with "translation-invariant" covariance

In physics literature, the covariance of a Gaussian measure $\mu$ on a function space is denoted as $C(x,y)$. Moreover, they say that if the covariance is translation-invariant, then actually $C(x,y)=\...
Isaac's user avatar
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2 answers
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Does strong stochastic ordering exist?

For two probability measure $\mu$ and $\nu$ on $\mathbb{R}$, we call $\mu$ is stochastically smaller than $\nu$ (i.e., $\mu\leq\nu$) , if $\int f \, d\mu\leq\int f \, d\nu$ for any nonnegative bounded ...
Jinxiang Yao's user avatar
2 votes
2 answers
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Limit of a integral whose integrand diverges under the limit

I am trying to simplify the following limit of integral where $\mu$ is given: $$p(y) = \lim_{\sigma \to 0} \int_{\mathbb R} |x| \cdot \frac{1}{\sqrt{2\pi\sigma^2} } e^{-\frac{1}{2\sigma^2} (xy - \mu)^...
user482401's user avatar
2 votes
0 answers
112 views

Product of marginals absolutely continuous with respect to a Borel probability measure

Let $\mu$ be a Borel probability measure on $\Bbb{R}^{m+n}=\Bbb{R}^m\times\Bbb{R}^n$. Consider its marginal measures $\mu_1(A):=\mu(A\times\Bbb{R}^n)\, (A\in\mathcal{B}(\Bbb{R}^m))$ and $\mu_2(B):=\mu(...
KhashF's user avatar
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3 votes
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An order statistics problem with some interesting geometry

Let $a_n$ be a given sequence of positive numbers, and $X_n$ a sequence of independent random variables with each $X_n$ uniformly distributed on $[0, a_n]$. Question: Let $N \geq 2$ be an arbitrary ...
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Asymptotic distribution of L infinity norm of Gaussian random vector

Let $\mathbf{X}_n = (X_{n,1}, \ldots, X_{n,n})$ be a $n$-dimensional random vector with $N_n( \mathbf{0}_n, \boldsymbol{\Sigma}_n )$ distribution. The asymptotic distribution of the $L_\infty$-norm of ...
joy's user avatar
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1 answer
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On the convergence in total variation

$\newcommand\R{\mathbb R}$For a probability measure $\mu$ over $\R^2$ and a unit vector $u\in\R^2$, let $\mu^u$ denote the pushforward of $\mu$ under the projection map $\R^2\ni x\mapsto u\cdot x\in\R$...
Iosif Pinelis's user avatar
7 votes
0 answers
211 views

Projected polar chessboard measure convergence in total variation?

$\newcommand\R{\mathbb R}\newcommand\C{\mathbb C}\newcommand\ga{\gamma}$For natural $n$, let $E_n$ be the set of all points in $\R^2$ with "polar coordinates" $(r,t)$ in the set $$F_n:=\...
Iosif Pinelis's user avatar
1 vote
1 answer
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Does convergence of Radon transforms of a sequence of probability distributions implies convergence of the distributions themselves?

Let $P_1,P_2,\ldots $ be a sequence of absolutely continuous probability measures on $\mathbb R^n$, and let $f_j:\mathbb R^n\to\mathbb R$ be their PDFs. Assume that $\operatorname{E}P_j = 0$ and $\...
Misha's user avatar
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2 answers
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Can a non integrable random variable satisfy a strong law of large numbers principle?

Given a random variable $X$, we denote by $X_1, X_2, \dots$ a sequence of iid copies of $X$. Question: Does there exist a random variable $X$ with $\mathbb E[X^+] = \mathbb E[X^-] = +\infty$, but $$\...
Nate River's user avatar
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Sufficient condition for linear interpolation of a 3D array to be a cumulative distribution function?

Let $z[i, j, k]\in [0, 1], 0\leq i, j, k \leq n$ be an array satisfying: $z[i, j, k]\geq z[i-1, j, k],z[i, j-1, k], z[i, j, k-1] $ $z[i, j, k]=0$ if $i=0$ or $j=0$ or $k=0$ $z[n, n, n]=1$. Now ...
AspiringMat's user avatar
1 vote
1 answer
119 views

Bound the distance between two vectors on the probability simplex

Let $a,b$ be two vectors with strictly positive elements and $\delta = 1 - \frac{\langle a,b \rangle}{\|a\|\|b\|}$. Bound the following optimization problem as a function of $\delta$ $$\sup_{x>0} \...
good bandit's user avatar
1 vote
0 answers
112 views

Continuity of a minimizing measure w.r.t a parameter

Let $V_t(x)=x^2+t\phi(x)$ where $t>0$ and $\phi\in C^\infty_c(\mathbb{R})$. My question is what can be said about the continuity of the (unique) minimizer (among probability measures) of the ...
BlueCharlie's user avatar
1 vote
0 answers
62 views

Random matrix theory: accounting for mean

Assume a random matrix, denoted as $X$, which is an $n$ by $T$ matrix, $T\geq n$. While I understand the typical scenario where the random variables $X_{ij}$ are sampled from a $\mathcal{N}(0,\sigma_{...
Pooja Algikar's user avatar
3 votes
0 answers
120 views

Extracting moments of $\max(X_1,\ldots,X_k)$ from asymptotic behavior of $\mathbb{E}[(X_1^n+\cdots+X_k^n)^m]$

For fixed $k$ suppose we have $X_1,\ldots,X_k$ non-negative random variables with density functions. Setting a): We know $\mathbb{E}[(X_1^n+\cdots+X_k^n)^m]$ exactly for any integers $n,m \in \mathbb{...
Tardis's user avatar
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4 votes
1 answer
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Probability to return to the origin for a uniform random walk

Consider a uniform random walk on $\mathbb{R}$, with stepsize chosen uniformly from the interval $(-1,1)$. The random walk start at $x=0$. Denote by $\rho_p dx$ the probability that the random walk ...
Carlo Beenakker's user avatar
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2 answers
201 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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2 votes
1 answer
218 views

Tests for determining membership of exponential family

Provided an arbitrary random variable $X$ with probability density/mass function $f$, are there any tests to determine if $f$ forms an exponential family? Certainly, if $f$ can be written in the form $...
Aaron Hendrickson's user avatar
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1 answer
72 views

Is the $2$-point function translation invariant for general Gaussian meaures?

Let us consider the real Hilbert space $H:=L^2\bigl(\mathbb{R}^n, \mathbb{R}^n\bigr)$ and "any" centered Gaussian measure $d\mu$ on it. Next, denote a generic element of $H$ by the column ...
Isaac's user avatar
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1 answer
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Joint distribution of randomly permuted Poisson random variables

Let $U_1, ..., U_n$ be Poisson random variables with rates $ \lambda_1, ..., \lambda_n$ such that $\lambda =\sum_i \lambda_i = O(1)$ (i.e the sum of the rates is bounded). Suppose we have $n$ buckets. ...
AspiringMat's user avatar
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1 answer
86 views

Where does this coupling result use independence when bounding total variational distance?

I am reading this paper, which gives the following coupling result: Throughout this, I'll assume the dimension $k$ is clear. Let $e_i$ be the $i$-th basis in the $k$ dimensional standard basis. A $k$ ...
AspiringMat's user avatar
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0 answers
44 views

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
3 votes
1 answer
167 views

Trace of product of two Wishart matrices

Let $A,B$ be two independent complex Wishart matrices, $A,B\sim CW_p(\mathbf{I},n)$, that is $A=\frac1n GG^\dagger$& $B=\frac1n QQ^\dagger$ where $G$ and $Q$ are independent $p\times n$ complex ...
Shadumu's user avatar
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1 vote
1 answer
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Asymptotic scaling of mean and variance for the norm of random vector (non gaussian components)

The norm of a vector whoose components $X_i$ are normally distributed follows the Non-central chi distribution and it can be shown that, increasing the number of components $k$ (i.e. the dimension of ...
user1172131's user avatar
3 votes
0 answers
156 views

Asymptotic distribution of the extreme, standardized order statistics of uniform distribution?

Let $\{U_{k, n}\}_{k=1}^n$, denote the order statistics of a sample of $n$ iid uniform $[0, 1]$ variates. Note that, marginally $U_{k, n}$ is distributed $\mathrm{Beta}(k, n+1 -k)$. Therefore, let us ...
Drew Brady's user avatar
1 vote
1 answer
125 views

Maximal entropy distribution on three variables knowing its marginals on any two

Observation 0: Given a finite set $X$, the probability distribution on $X$ with highest entropy is the uniform one. This is well known. Observation 1: Given two finite sets $X,Y$ and two probability ...
Gro-Tsen's user avatar
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Random variables and marked point processes

Let $(\Omega,\mathcal{A},\mu)$ be a probability space. Suppose that $f:\Omega\times\mathbb{R}\to \mathbb{R}$ is a given function (that can be continuous in its second entry). Is it possible to ...
Jcnak's user avatar
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2 votes
2 answers
210 views

Asymptotic scaling of mean and variance for non-central chi distribution

Define $Y \equiv \sqrt{\sum_{i=1}^k(\frac{ X_i}{\sigma_i})^2}$, with $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$ and independents. It is known that $Y$ is distributed as a non-central chi (Noncentral ...
user1172131's user avatar
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109 views

Question on the inverse Mellin transform $p(x)=\mathcal{M}_s^{-1}\left[-\xi(s)\,\frac{\zeta'(s)}{s\,\zeta(s)^2}\right]\left(\frac{1}{x}\right)$

Consider the function $$p(x)=\underset{K\to\infty}{\text{lim}}\left(\sum\limits_{k=1}^K \Lambda(k) \left(\frac{2 \pi k^2}{x^2}-1\right) e^{-\frac{\pi k^2}{x^2}}\right)\tag{1}$$ where $$P(s)=s\, \...
Steven Clark's user avatar
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1 vote
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Monotone likelihood ratio of convolved power function kernel, $p\ge 3$

It was shown in a previous answer that for: $f(x)=|x|^p$, $\;x\in \mathbb{R}$, $\;p>2$, defining the density: $$p(\phi;\theta) = \Big(f\big(\hspace{-1pt}\cos(\phi-\theta)\big) - f\big( \hspace{-1pt}...
japalmer's user avatar
  • 141
2 votes
1 answer
177 views

Gaussian expectation restricted to a convex polytope

Let $X$ be a Gaussian vector in $\mathbb{R}^n$ with $\mathbb{E}[X]=0$ and $\mathbb{E}[X X^\intercal]=I_n$. Let $\mathbf{S}$ be a convex polytope in $\mathbb{R}^n$ defined as the intersection of $m$ $(...
Ye He's user avatar
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1 vote
1 answer
87 views

Scale convolution decomposition of a density

Is it possible to decompose the density: $$p(x) = \frac{8}{\,\pi^2} \frac{x^3\tanh(x)}{\cosh^2(x)},\quad x>0$$ into a scale convolution of two non-negative densities: $p(x) = \int_0^{\infty} \xi^{-...
japalmer's user avatar
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2 votes
0 answers
240 views

Why is it impossible to create a numerically balanced die with more than 120 sides?

I allow myself to contact you as a mathematics enthusiast. I have recently been intrigued by the concept of balance in dice and the assertion that it would be impossible to create a numerically ...
Matthieu Nauly's user avatar

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