Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

3
votes
1answer
76 views

Distribution of largest entry in a random vector

If we have a random unit vector on $\mathbb{C}^n$, drawn from the Fubini-Study metric, the marginal distribution of the squared absolute values of each of the coefficients in the vector is given by a ...
0
votes
0answers
46 views

Optimization of generalized Fisher information over space of probability measures

Assume we already have $p(\theta)$ apriori desnsity function about unknown parameter $\theta$, and we consider an optimization problem as follows: \begin{equation} \max_{P \in \mathcal{M}} \int_{\...
-1
votes
0answers
35 views

Compounding two Dirichlet distributions

If $\pmb y|\pmb x\sim\text{Dir}(\alpha\pmb x)$ and $\pmb z|\pmb y\sim\text{Dir}(\beta\pmb y)$, then what can be said about the distribution of $\pmb z|\pmb x$? For example, is it possible to derive ...
2
votes
0answers
80 views

Trace of Symmetric matrices in fixed rank

I am solving some problem related to symmetric matrices over a finite field $\mathbb{F}_q$ and I am stuck at the following problem: For every $a\in\mathbb{F}_q $, let $S_a(t,m)$ be the set of all $m\...
3
votes
1answer
335 views

Earth movers distance (EMD) between two multivariate normals. Is it negative definite distance?

I was looking at the closed form formula for 2-Wassersteins distance for multivariate normal distribution on Wikipedia. https://en.wikipedia.org/wiki/Wasserstein_metric#Normal_distributions It has a ...
1
vote
1answer
46 views

Probabilistic meaning of maximal rectangle under probability distribution function graph

Let $\xi$ be a random variable with $p(x)$ density function, which is like a normal distribution (i.e. $p(x)$ is increasing on $(-\infty,0]$ and decreasing on $[0,\infty)$). Denote by $s$ the maximum ...
0
votes
0answers
26 views

proving separable structure for positive numbers

I want to prove that if I have $$q_{ik}q_{lj}=q_{lk}q_{ij}$$ for any 4 indices $i,j,k,l$ such that $i\neq l$ and $k\neq j$, and $$q_{ik},q_{lj},q_{lk},q_{ij}\geq 0, \forall i,j,k,l$$ it follows ...
5
votes
1answer
132 views

Moments of maximum of independent Gaussian random variables

Let $X = (X_1, \ldots, X_d) \in \mathbb{R}^d$ be a mean-zero Gaussian random vector with identity covariance matrix. Are there upper bounds for $$E \left(\|X\|_{\infty}^k \right)$$ for $k=1, \ldots, ...
1
vote
0answers
52 views

Bounding quantiles of the noncentral chi distribution

I need to bound the empirical quantiles for a noncentral chi distribution (not chi-squared) $\chi_\nu(\lambda)$, where $\nu$ is the number of degrees of freedom and $\lambda$ the non-centrality ...
0
votes
0answers
62 views

In Wasserstein, what is the relationship between $W_2 (\widehat{\mathbb{P}}_{N},\mathbb{P})$ and $W_2 (\widehat{\mathbb{P}}_{N}^{x},\mathbb{P}^{x})$?

Before presenting my question (which I already formulate in the title of this post) is important to establish the context of my problem: Definition: The $p$-Wasserstein metric $W_{p}(\mu,\nu)$ ...
1
vote
1answer
74 views

One question about compensated Poisson process

Let $N$ be a Poisson process with parameter $\lambda$, that is, for $a>b\geq0$, there is $$P[N(a,b)=k]=\frac{((a-b)\lambda)^k}{k!}e^{-(a-b)\lambda}.$$ Now denote $N_t=N[0,t)$ and define $$ M_t=N_t-...
-1
votes
2answers
152 views

Bounded difference functions and sub-Gaussian random variables

We have the following standard theorem : Let $X$ be some set and $g : X^n \rightarrow \mathbb{R}$ be a measurable function such that it satisfies the ``bounded difference property" i.e $\exists$ $\{...
3
votes
1answer
182 views

Symmetry in the triangular distribution

A continuous function vanishes on $(-\infty,a]$ and on $[c,\infty),$ its graph is a straight line on the interval $[a,b]$ and another straight line on $[b,c],$ and its integral is $1.$ The mean of ...
1
vote
1answer
95 views

Ratio of perfectly correlated gaussian distributions

Let $M$ be a positive definite matrix and let $w \in S^{d-1}$ be a unit vector uniformly distributed over the sphere. I want to understand the distribution of the quadratic form $\frac{w^T M^3 w}{w^T ...
4
votes
0answers
110 views

Under what conditions is conditional expectation a bijective operator of distributions

Suppose that we have two independent random variables $V$ and $W$ over $\mathbb{R}$. Suppose that $W$ has a probability density with respect to the Lebesgue measure. My question: Can we find ...
1
vote
1answer
113 views

Is this function measurable?

Let $(B, \Sigma_B)$ and $(C, \Sigma_C)$ be standard Borel spaces and let $\mu$ be a sub-probability measure on $C$. Given $Y\in \Sigma_{B\times C}$, I would like to use the following function: $$ f:...
1
vote
1answer
76 views

Calculate Average and Correlation of WSS Random Processes

Given two stochastic processes, $X[n]$ and $Y[n]$, both being WSS (wide state stationary) and independents. What would be the Average and Autocorrelation function of $Z[n] = Y[n] X[n]$? Is the ...
2
votes
1answer
99 views

$p$-th moment of complex Gaussian random variable

Let $1<p<2.$ Let $G$ be a complex Gaussian random variable. then what is the value of $\mathbb{E}[|G|^p]$ ? The symbol $\mathbb{E}$ denotes the expectation of a random variable.
0
votes
1answer
158 views

Berry-Esseen type theorem for Monotonic independence

The central limit theorem states that, under certain circumstances, the probability distribution of the scaled mean of a random sample converges to a normal distribution as the sample size increases ...
6
votes
0answers
262 views

Calculate the expectation of the maximum of averaged random walks

Let $X_1, X_2, \ldots$ be iid random variables with bounded second moment. The question is to calculate the exact value of $$\mathbb{E} \max_{1 \le j < \infty} \frac{X_1 + \cdots + X_j}{j}.$$ Is ...
1
vote
0answers
139 views

References for the theory of Hadamard functions and compositions of random vectors

Recently, I fell in love with the pointwise/elementwise/componentwise/Hadamard/Schur functions and compositions of random vectors such as Hadamard squares and products of random vectors. Here is one ...
3
votes
0answers
83 views

Probability distributions with all positive cumulants

Is there a term for a distribution with all cumulants positive (or nonnegative)?
4
votes
0answers
79 views

Asymptotics of the joint pdf of two sums of powers of independent $\mathcal U(0,1)$ random variables

As a warm-up in words: The sum of twelve uniform random variables is a classic approximation to a normal distribution. What is the joint pdf for the sum of their cubes and the sum of their fourth ...
2
votes
0answers
60 views

“Optimal” local limit theorems for densities vanishing at zero

Consider a nonnegative stable distribution with a density that vanishes at zero, such as $$f(t)=\frac{e^{-1/2t}}{\sqrt{2\pi t^3}},\qquad t\geq0.$$ Suppose (for simplicity) that we have i.i.d copies $(...
2
votes
1answer
404 views

Design a Galton Board to simulate a uniform distribution

This link http://mathworld.wolfram.com/GaltonBoard.html suggests that a certain specific placement of pegs can be used to simulate a binomial or a normal distribution. Is there a specific peg ...
3
votes
1answer
265 views

Normal approximation to the pointwise/Hadamard/Schur product of two multivariate Gaussian/normal random variables

Let $X \sim \mathcal{N}\left( {{\mu _x},\sigma _x^2} \right)$ and $Y \sim \mathcal{N}\left( {{\mu _y},\sigma _y^2} \right)$ be two univariate and independent Gaussian/normal random variables and let $...
6
votes
2answers
173 views

maximal distance of nearby iid unifrom random variables

Question: Let $X_1, \ldots ,X_n$ be $n$ iid uniformly distributed random variables, i.e., $X_j \sim \mathcal{U}(0,1)$ for each $j=1,\ldots ,n$. What is the PDF of the maximal distance between to ...
4
votes
0answers
207 views

Minimization over a convex function of equal vs unequal success probabilities of Bernoulli random variables

Let $U_1,U_2,\ldots,U_n$ be $n\geq 2$ mutually independent Bernoulli random variables. There are two cases of interest: $1.$ The random variables $U_1,U_2,\ldots,U_n$ are identically distributed; $...
3
votes
2answers
194 views

Log-concavity of the maximum of gaussians

Let $Z_1,\ldots, Z_n$ be independent standard gaussian random variables. Is it true that $X=\max\{Z_1,\ldots,Z_n\}$ has a log-concave distribution function?
2
votes
1answer
117 views

Minimising the f-divergence to a conditional probability constraint

Let $P$ be a probability distribution and let $A$ and $B$ be some events, and suppose that we want to minimise an $f$-divergence between $P$ and the set of all distributions $Q$ that satisfy that ...
3
votes
1answer
366 views

Kullback–Leibler divergence of product distributions

Say the KL divergence between two distributions $A$ and $B$ is $\varepsilon$. Can we give bounds, or a precise computation, of the KL divergence between $A^k$ and $B^k$ (the product distributions)?
2
votes
1answer
209 views

CLT for Martingales

I posted this question originally in math stack exchange, but I got no answer. (https://math.stackexchange.com/questions/2604591/clt-for-martingales) In wikipedia, there is a version of a CLT for ...
5
votes
0answers
170 views

Total Variation distance of polynomials of Bernoulli R.V.s

Let $X_i, Y_i$ be i.i.d Bernoulli $0/1$ random variables with $\mathbb{E}[X_i] = p$ and $\mathbb{E}[Y_i] = q$. Let \begin{align*} X &= X_1 X_2 + Χ_2 Χ_3 + \ldots +X_{n-2} X_{n-1}+ X_{n-1} X_n\\...
5
votes
1answer
104 views

CLT for Bernoulli RV with negative correlation

Suppose $X_1,X_2,...$ are Bernoulli random variables with $P(X_i=1)=p_i$ and $X_i$ have negative correlation. Is there a CLT in this case, i.e. does $\frac{Z_n-(\Sigma^n_{i=1}p_i)}{\sqrt{n}}$ converge ...
1
vote
1answer
300 views

Independence of stochastic processes [closed]

Suppose that $(X_t)$ and $(Y_t)$ are stochastic processes defined on the same probability space whose sample paths belong to some Hilbert space $K$ (or more generally some function space). We may view ...
5
votes
1answer
162 views

uniquely determining a distribution using moments

Let $A$ be a parametric family of probability distributions that include all distributions in the form of $\phi(X)$ where $X\sim\mathcal{N}(0,\mathbf{I})$ is jointly Gaussian and $\phi:\mathbb{R}^d\to ...
2
votes
0answers
85 views

On a random matrix construction

Given a symmetric matrix $M\in\Bbb Z^{n\times n}$ or rank $r$ with absolute value of any entry bound by $2^{b^2-1}-1$ and maximum eigenvalue at most $\lambda$. We consider the set $\mathcal T_b$ of $\...
4
votes
1answer
418 views

How to measure distribution of high-dimensional data

I have to methods of projecting random samples in $\mathbb{R}^n$ onto a manifold defined by $C(q)=0$, which is a lower-dimensional subset. Now, samples in $\mathbb{R}^n$ are uniformly distributed. ...
6
votes
1answer
154 views

Martingale version of Bernstein-type inequality for (slightly) heavy-tailed distributions?

It is known that for sub-exponentially distributed martingale difference sequence, the following Bernstein-type inequality holds: $$ ℙ\left(\left| \sum_{i=1}^N a_i X_i \right| \ge t \right) \le 2\...
2
votes
1answer
296 views

Distribution of ratio between complex Gaussian and Chi-square R.V.s

What would be the distribution (p.d.f.) of the following ratio? $$z = \frac{x_{1}}{|x_{1}|^2 + |x_{2}|^2 + ... + |x_{M}|^2}$$ where $x_{i} \sim \mathcal{CN}(0,a), \forall i$ and $a > 1$. As can ...
2
votes
1answer
250 views

Cantelli's inequality: the original source

Does anyone know where and when Cantelli's inequality was originally published? Strangely enough, I have not been able to find this information online.
2
votes
0answers
239 views

Calculating Wasserstein's distance between an empirical distribution and a combination of normal distributions

Context of the problem Let $\xi$ be a random variable (with real value) with support $\Xi=\mathbb{R}$ and $\xi_{1},\ldots,\xi_{N}$ be a sample of $\xi$. We consider the empirical probability $$\...
0
votes
0answers
92 views

Lower bound for the probability that $X=\omega\left(\mathbb E[X]\right)$ for $X\sim Bin(n,p)$

Let $X\sim Bin(n,p)$ be a binomial variable and let $\delta\in (0,1)$. I'm looking for a lower bound of the form $\Pr[X > f(\delta)] \ge \delta$. Specifically, if $\delta,p=o(1)$ are not ...
3
votes
1answer
191 views

Asymptotic form of pdf of Escape Time of arithmetic fBm

I am trying to apply the Girsanov formula and Doobs optional sampling theorem to obtain an asymptotic form of first passage density of an fbm process with drift, but the answer i am getting seems ...
3
votes
1answer
148 views

Bounding the “spikiness” of a probability distribution

Are there any well-known conditions that guarantee that a probability distribution isn't too "spiky"? I ask this question because I am interested in the families of probability distributions $f(x)$ ...
3
votes
1answer
131 views

Large deviations for integrands

I am a physicist caught in the following situation: I have two probability measures $\mathbb{P}_1$ and $\mathbb{P}_2$ and have to deal with the following integral where $X_i$ are random iid: $$\int_{...
0
votes
0answers
73 views

Probability of getting through with a phone-call

Alice is quite popular. She gets called on her cell-phone in a Poisson$(\lambda)$ manner. She answers her calls when possible, and ignores them when in the middle of conversation. Since you know her ...
1
vote
0answers
89 views

Higher order derivative of negative power of cosine function

This is a question I encountered in my own research on Generalized Hyperbolic Secant (GHS) distributions. It is known that the Laplace transform of the basis measure for this family is $$L\left( \...
2
votes
0answers
109 views

Modified Wigner semicircle law

The Wigner semicircle law states that for a random GOE-matrix $M^N \in \mathbb{R}^{N \times N}$ in the $N \rightarrow \infty$ limit for any $f \in C^b(\mathbb{R})$ $$\lim_{N \rightarrow \infty}\frac{...
3
votes
1answer
90 views

integral involving hypergeometric function of matrix argument

This conjecture comes from an observation on simulations of the matrix variate noncentral Beta distribution (similar to this observation, but I open a new question because yet I'm not sure it is ...