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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Random permutations constructed via randomly chosen transpositions

Given positive integers $k$ and $n$, we define the probability distribution $p_{n,k}$ on $S_n$ as: $$ p_{n,k}(\sigma):=\frac{\#\{(\tau_1,\dots,\tau_s)\mid \sigma=\tau_1\dots\tau_s, \text{ each }\tau_i\...
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Optimal Monte Carlo Trace Estimator

For a psd real symmetric $d\times d$ matrix $A$ and a function $f: \mathbb{R}^d \to \mathbb{R}$, with $f(x) := x^T A x$ we have that with $p(x) = \mathcal{N}(0_d, I_d)$ (i.e. standard multivariate ...
Sebastian Nowozin's user avatar
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Approximating a probability density with a point set

Let $f$ be a "nice" probability density on $\mathbb{R}^2$, let $p=1/k$ for some fixed positive integer $k$, and let $\epsilon>0$. Are there any known statements of the following form? &...
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Prove or disprove a mutual information inequality

I have $n$ IID Bernoulli random variables denoted by $X_1,X_2,\ldots X_n$ with parameter $p$. I am interested in knowing if the following inequality involving mutual information holds : $\boxed{\max_{...
wanderer's user avatar
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Distribution of unbiased estimator of covariance matrix with missing values

Initial setup Assuming $X_1, ..., X_n \in \mathbb{R}^m$ are iid, sampled from $\mathcal{N}(\mu, V)$, one can define the estimators for the sample mean $\hat{\mu} = \frac{1}{n} := X^T 1_n$, and sample ...
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Concentration of the norm of subGaussian random vectors

I will use the same notation and definitions in High Dimensional Probability, by Roman Vershynin. I have a sub-Gaussian vector $y$, in $\mathbb{R}^n$ and sub-Gaussian norm $C$ non dependent on $n$. I ...
SiMohani's user avatar
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Pagerank Markov chain reductions

In short: if a Markov chain models a (generalized) pagerank, is it always possible to remove any of its state and obtain a Markov chain that models a pagerank close to the initial one? Full details. ...
Matthieu Latapy's user avatar
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A slight generalization of Skorokhod's representation theorem

Let $f:\mathbb{R}^p\rightarrow\mathbb{R}^q$ $(p,q\geq 1)$ be a continuous function and $(X_n)_{n\geq 1}$ a sequence of random values on $\mathbb{R}^p$ such that $f(X_n)$ converges in law to a random ...
G. Panel's user avatar
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Log-concave probability measure with slowest decay

Let $X$ be a real valued random variable with log-concave distribution $\mu$. For each $x \in {\mathbb R}$, let $$ \phi_\mu(x)=\min\limits_{c\in{\mathbb R}}E[e^{c(X-x)}] $$ be the minimal value of the ...
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Simplify Kantorovich–Rubinstein duality when distributions share a common marginal

Consider the product of two metric spaces $X\times Y$, and two probability distributions $\mu$ and $\nu$ on this product space. By the Kantorovich-Rubinstein duality, I can write the Wasserstein-1-...
joemrt's user avatar
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Maximum entropy distribution in the hyperbolic plane with given "mean" and "variance"

On the Euclidean sphere, by defining suitable analogs of the mean and variance in terms of the first circular moment, it can be shown that the von Mises distribution is the maximum entropy ...
popstack's user avatar
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Random sets of points and hyperplanes in high dimensions

We are given a set $X$ of $n$ points $\mathbf{x}_1, \mathbf{x}_2, \ldots, \in\mathbb{R}^d$ selected uniformly at random from the unit origin-centered ball $\mathcal{B}^{d}$. Consider the random ...
Penelope Benenati's user avatar
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Concentration inequalities for sets

Assume that we have a random set $B$ which is constructed by selecting elements from $U = \{ X_1, \dots, X_n \}$ where $X_i$ are independent samples from Gaussians with means $\mu_i$ and variances $\...
bolzano's user avatar
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General definition for $k$-dependence of a family of sub-$\sigma$-algebra

If $(\Omega,\mathcal{F},\mathbb{P})$ is a probability space, is there a general definition for the "$k$-dependence" of an arbitrary family $(\mathcal{F}_i)_{i \in I}$ of sub-$\sigma$-algebra ...
john's user avatar
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The CDF of determinant of wishart distribution

Assuming $b>0$, $\mathbf{A} \in \mathbb{C}^{m\times n}$ with $m\leq n$ and each element of $\mathbf{A}$ is i.i.d. $\mathcal{CN}(0,1)$ distributed, how to obtain $\mathbb{P} \left[ \det\left( \...
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Finding an optimal strategy for a combinatorial sequential game

We are given a set $\{p_1, p_2, \ldots, p_n\}$ of players and a set of $\{\ell_1, \ell_2, \ldots, \ell_n\}$ of locations, where $n\in\mathbb{N}$. Each location can be either free or occupied, and each ...
Let101's user avatar
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Adding elements in a list *in expectation*

Suppose $𝐿_1,…,𝐿_𝑘$ are lists with $n$ elements each. We use a fully independent hash function ℎ to compute a value for each element of each list. (We suppose the hash function returns a value ...
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What are some beginner's references on algebraically structured (statistical) models, and their connection with group actions and Fourier transform?

I asked this question on Cross Validated a few days ago, but didn't really get a favorable response, so asking here to see if I get any. I'm looking at the description of a short-term position in ...
Stat_math's user avatar
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approximate the square of 2-norm distance between binary distributions with high probability

Suppsose we take $m$ samples from a Bernoulli distribution with probability $p$, and $m$ samples from another probability distribution with probability $q$. We want to calculate a statistic $x$ from ...
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An inequality of KL Divergence for two different distributions passing through a same channel

Let $X$ be a random variable which takes values in $\mathcal{X}$. Assume that we pass $X$ through two independent conditional pdf $p_{X_1|X}$ and $p_{X_2|X}$ and choose $X_1$ with probability $\lambda$...
Math_Y's user avatar
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Average of random variables is "more log-concave"

Problem (1) Suppose $\phi_i\in [0,\pi/2]$ are drawn uniformly for $1\le i\le n$, and $\sum_{i=1}^n w_i=1$, $w_i\ge 0$. Show that the pdf $p_1$ of the random variable $$\phi = \sin^{-1}\left(\sqrt{\...
Holden Lee's user avatar
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Less regular version of the Gaussian free field

One can define (continuous) Gaussian free field as follows: one can consider some orthonormal basis $(\psi_k)_{k=1}^{\infty}$ in the Sobolev space $H^1(\Omega)$ (here $\Omega \subset \mathbb{R}^d$) ...
truebaran's user avatar
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Do there exist iid random variables $X$, $Y$ with countable support such that $X + Y$ and $X Y$ are also distributed with the same parameterisation?

This is perhaps a well-known result and I'd appreciate a reference if that's the case. Let $X$, $Y$ be iid random variables with support $S \subset \mathbb{Z}$ and let $$P(X = x) = f(x, \theta)$$ ...
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Monotone coupling between "two-sided Gumbel" distributions

I am interested in finding a monotone coupling between two random variables. Let $\alpha_1>\alpha_2$, $b<a$. Define the following two (non-normalized) densities on the whole real line: \begin{...
Leonid Petrov's user avatar
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How fast does a sum of Bernoulli distributions (of different parameters) decrease after its mean?

Let $X=\sum_{i=1}^nX_i$, where each $X_i$ is a random variable following a Bernoulli distribution of parameter $p_i$. All $X_i$ are independent, and for all $i$, $p_i<p$ for some small $p$. I'm ...
Ted's user avatar
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Infinite divisiblity of log-normals

What is the law of a piece of a log-normal distribution? We know that log-normals are infinitely divisible. What would be the law of a root of log-normal? More specifically, suppose that $X$ is a ...
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Uniqueness of martingale problem for Levy type operator

Consider the following Levy type operator: $$ L_t\varphi(x)=\int_{R^d}\big[\varphi(x+z)-\varphi(x)-1_{|z|\leq 1}z\cdot\nabla\varphi(x)\big]\kappa(x,z)\nu(dz),\quad\forall \varphi\in C_c^2(R^d), $$ ...
Wenguang Zhao's user avatar
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On the difference of conditional differential entropy of two correlated random variables

Problem Definition Let $\mathbf{G}$ and $\mathbf{S}$ be jointly distributed random variables where $\mathbf{S}$ is continuous and is related to $\mathbf{G}$ through a conditional pdf $f(s|g)$ defined ...
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An inequality regarding centered Bernoulli random variables

Let $\left\{V_1,\ldots,V_n\right\}$ be a set of (possibly dependent) identically distributed Bernoulli random variables, with $$ p = \mathbb{P}\left(V_i=1\right) = 1-\mathbb{P}\left(V_i=0\right),\quad ...
Student's user avatar
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Distribution of a linear pure-birth process' integral

I stumbled across the following random variable, defined as the integral of a linear pure-birth process i.e. a Yule process: $$ Z_t = \mathbb{E}\bigg[\int_0^t Y_s ds \bigg| Y_t=k , Y_0=1\bigg] $$ ...
Gabriel's user avatar
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Statistics of perfect matching and incremental perfect matchings in bipartite planar graphs?

Planar graph permanent can be reduced to determinants and so statistics should be amenable. Pick a uniformly random bipartite planar graph $G$ with $n$ vertices of each color and choose new additional ...
Turbo's user avatar
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Distribution Functions in Poisson Process

We consider definition of Poisson processes that satisfy Condition 0 and 1 according to Billingsley section 23 . How find out the densities of $A_t , B_t , L_t$ defined as in problems section of ...
John's user avatar
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Given a set of marginals, what is the largest support of a distribution satisfying these?

Given a random variable $X$ with support over $\{0,1\}^I$, we can define the marginal distribution on the bits indexed by $A \subseteq I$ by $Pr(X_A = x_A) = \sum_{x \in \{0,1\}^{I - A}} Pr(X = x \cup ...
Samuel Schlesinger's user avatar
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If $X^n$ is a sequence of càdlàg processes whose FDDs converge to a continous process $X$, does $X^n$ converge to $X$ in the Skorohod topology?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $E$ be a complete locally compact separable metric space, $(X^n_t)_{t\ge0}$ be an $E$-valued càdlàg process on $(\Omega,\mathcal A,\...
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Reference Request: Total Variation Between Dependent and Independent Bernoulli Processes

Let $X$ be a random variable taking values in $\{0,1\}^n$ with the following distribution. For each coordinate $i$, we have $p_i = P(X_i = 1) = c/\sqrt n$, where $c$ is a (very small) constant. ...
Sam OT's user avatar
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Distribution of weighted sum of non-central chi-squared r.v.'s: log-concave?

Let $n\geq 2$, and $X_1,\dots,X_n$ be independent non-central r.v.'s, where $X_i \sim \chi^2(\delta_i)$; and $w_1,\dots, w_n > 0$. Letting $$X \stackrel{\rm def}{=} \sum_{i=1}^n w_i X_i$$ is it ...
Clement C.'s user avatar
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Random walk and comparing sums of Exponential random variables

Let $\sigma$ be the time a nearest neighbor random walk started at 1 that has probability $p>1/2$ of moving left reaches $0$. Let $\sigma'$ be an independent copy of $\sigma$. Let $(X_k)_1^\infty$ ...
Matthew Junge's user avatar
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90 views

What are the moments of Kolmogorov Complexity for a Random Variable?

Given a random variable $X$ distributed under some computable distribution $P$ we have, $$0 \le E[K(X)] - H(P) \le K(P)$$ Where $H(P)$ is the entropy of $P$. I tried using Hoeffding concentration ...
Zachary W. Robertson's user avatar
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0 answers
255 views

Schilder's theorem for brownian bridges

I am really not a probabilist and I apologize if my question is too naive or not appropriate, please feel free to migrate to SE. A bit of context: usually, Schilder's theorem tells us that the ...
leo monsaingeon's user avatar
2 votes
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223 views

Singular values of random matrices with inhomogeneous variances

If $X$ is a random rectangular matrix with independent identically distributed entries of zero mean and equal variance, then as $X$ gets big its singular values tend to a Marchenko-Pastur distribution....
Neuromath's user avatar
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On the convergence problem of box counting for the Rössler attractor

So 5 month ago i posted this Question: Rössler attractor, Convergence of box counting to estimate the fractal dimension. Since then i have assumed, that the rate of convergence of $n(ɛ,n)$ (sum ...
Chopin's user avatar
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"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 $(...
user78370's user avatar
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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 $\...
Turbo's user avatar
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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 $$\...
matematicaActiva's user avatar
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122 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{...
user avatar
2 votes
0 answers
61 views

Existence and uniqueness of fixed point in generalized condition of triangular norm

Definition 1) A Menger space is defined as a triple $\left( S,F,T \right)$ where $S$ is a set , $F$ is a collection of distribution functions and $T$ is a triangular norm function $T:[ 0,1 ]\...
nastaran noorivatan's user avatar
2 votes
0 answers
56 views

Rate of $L_1$ loss in estmating density on $[0,1]$

Let $f$ be a density on $[0,1]$ and let $X_1,X_2,\ldots$ be $\textit{iid}$ $f$-distributed. Also, let $f_n$ denote the kernel density estimator, i.e. $$f_n(x) = \frac{1}{nh_n} \sum_{i=1}^n K\left(\...
Yannik's user avatar
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Moments of a Normal-Wishart distribution

Do known expressions exist for the moments of a gaussian-wishart (aka normal wishart) distribution? $$NW(\mu,K\mid\mu_0,\lambda_0, v, W) = \frac{|\lambda_0K|^{1/2}}{(2\pi)^{d/2}}e^{-0.5([\mu - \mu_0]...
ejlouw's user avatar
  • 121
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179 views

Splitting of a random variable

Can we characterize the laws of n-uple real random variables $(X_1...X_n)$ so that for any random variable Y which has the same law as the sum of the $X_i$ there exists a n tuple of functions $(f_i)_i$...
Raymond F.'s user avatar
2 votes
0 answers
80 views

Statistical properties from the projection on Hadamard matrix

I am using Hadamard matrix to generate the measurement matrix in compressive sensing for signal acquisition. Consider all the rows except the first row, which consists of all 1's, as a candidate for ...
Amit's user avatar
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