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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Projection of log-concave distribution on unit sphere surface

Let $\mathbf X : \Omega \to \mathbb R^d$ be a random vector following a zero mean, identity covariance log-concave distribution. Is there any known upper bound for the probability density function of $...
entechnic's user avatar
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Variance of the logarithm of the mixed Rademacher and complex Gaussian distribution

Consider the scenario where $X$ is a Rademacher random variable taking values $\{-1,+1\}$ with equal probability, and $Z$ is a complex Gaussian random variable with a mean of $0$ and a variance of $\...
Math_Y's user avatar
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Sample integral points in m-Ball

The problem I have is pretty simple, however I cannot find an answer. I need an efficient algorithm to sample integral points within an m-dimensional ball of radius r around the origin (Euclidean norm)...
N. P.'s user avatar
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Is it reasonable to consider the subgaussian property of the logarithm of the Gaussian pdf?

Let $Y$ denote a Gaussian random variable characterized by a mean $\mu$ and a variance $\sigma^2$. Consider $N$ independent and identically distributed (i.i.d.) copies of $Y$, denoted as $Y_1, Y_2, \...
Math_Y's user avatar
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Hoeffding's Lemma for bounded complex random variables?

If we have a real random variable $X$ such that $a\leq X\leq b$ almost surely, we can establish the following inequality: \begin{align} \mathbb{E}\left[\exp\Big(t(X-\mathbb{E}[X])\Big)\right]\leq\exp\...
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characteristic function and weak convergence

Let $F_n,G_n$ be with distribution functions with characteristic function $f_n,g_n$, if $f_n-g_n\rightarrow 0 \, a.e.$, then for each $f\in C_K$ \begin{equation}\int f\mathrm{d}F_n-\int f\mathrm{d}...
Sheng Wang's user avatar
5 votes
1 answer
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How can I sample uniformly from a citrus surface?

I want to sample from a Lemon surface uniformly. The equation of this surface is $$16(x^2+z^2)+(y-2)^3 y^3=0.$$ I have read the paper Stratified Sampling of 2-Manifolds . The method described in this ...
haotian yi's user avatar
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Simple anticoncentration bound for binomially distributed variable

The following question, which arose during my research, seems deceivingly simple to me, but I could not find any elegant and formal argument. For a binomially distributed variable $X \sim \text{Bin} \...
reservoir's user avatar
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Calculation of the difference of two Brownian bridges

I was told that the difference of two independent brownian bridge process is $\sqrt{2}$ times a brownian bridge process, i.e., $$B_{1t} - B_{2t} = \sqrt{2}B_t$$ where $B_{1t}$ and $B_{2t}$ are ...
John Smith's user avatar
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Methods for high-dimensional integration of joint PDFs over unbounded regions

I am currently grappling with an issue related to high-dimensional integration of joint Probability Density Functions (PDFs) over unbounded regions. The specifics of the problem are a bit intricate, ...
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Lipschitz property of the determinant

$\newcommand{\A}{\mathcal A}\newcommand{\Tr}{\operatorname{tr}}$For $c$ and $C$ such that $0<c<C<\infty$, let $\A_{d;c,C}$ denote the set of all symmetric positive-definite real $d\times d$ ...
Iosif Pinelis's user avatar
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Probability distribution for a Bayesian Update

I am struggling with a process like this: $$X_t=\begin{cases} \frac{\alpha\omega_t}{\alpha\omega_t+\beta(1-\omega_t)} & \text{with prob } p\\ \frac{(1-\alpha)\omega_t}{(1-\alpha)\omega_t+(1-\beta)(...
DreDev's user avatar
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$L_1$ norm concentration of an empirical distribution

Suppose we have one random variable $X$, whose sample space is $\mathbb{X}=\{x_1,x_2,\dots,x_m\}$, and the size of the sample space is $m$. We have $N$ i.i.d. samples from this distribution, and $x_i$ ...
white's user avatar
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An exercise on log-concave random variable on the real line

Let $X$ be a real random variable with log-concave density $f$. Assume that $E(X) =0$ and $E(X^2)=1$. Show that there is a universal (independent of $X$) constant $c>0$ such that: $$P(X\in[-1/2;0])\...
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Approximation to ratio distribution

Recently I was thinking if there is a way to do the following: assuming I have some sampled points of distribution $\mathcal{X}$ and distribution $\mathcal Z$ (whose MGF I do not have in closed form) ...
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How to augment dataset keeping its distribution? [closed]

I have a dataset with data points consisting of several tabular features and outputs. I want to generate a more extensive dataset maintaining the original distribution. Formally: Given a set of N ...
Alejandro Jimenez's user avatar
4 votes
2 answers
271 views

Effect of small change in probability distribution on error probability

Let $X$ be a random variable and $Y=f(X)$ where $f$ is a deterministic function. Moreover, assume that there exists a deterministic function $g(.)$ such that the following probability is small. \begin{...
Math_Y's user avatar
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Inequality for log-likelihood ratio

Let $ p, q $ be two probability densities on $ [0,1] $, strictly positive over $ (0,1) $. Let $ P $ be the cumulative function of $ p $, i.e., $ P(x) = \int_0^x p(x') \, \mathrm{d}x' $, $ x \in [0,1] $...
aleph's user avatar
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1 answer
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How to solve an optimization problem whose optimization variable is a function?

I would like to find an optimal probability density function (PDF) $f$. Given $b$, $$ \begin{array}{ll} \underset {f} {\text{minimize}} & C \\ \text{subject to} & 1 + \frac{b}{x} \displaystyle\...
Erik's user avatar
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Closed form volumes for intersecting modified cylinders

This question is somewhat related to the question Intersecting cylinders, but where the cylinders are now modified to orbifolds in the hypercube with singularities occurring at the vertices of the ...
53Demonslayer's user avatar
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What can we say about the order of convergence of a critical point of Gaussian mixture density to its limit when the parameter $h$ goes to $0?$

Density of Gaussian mixture with $n$ components is given by: $$f(x):=C \sum_{i=1}^{n}e^{-\frac{1}{2}||\frac{x-x_i}{h}||^2}, x_i \in \mathbb{R}^d, h > 0$$ where $C$ is a normalization constant ...
Learning math's user avatar
2 votes
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71 views

A complex problem involving densities (likelihood functions) and optimization

Consider the following autoregressive process with normal errors: \begin{equation}\label{7YlUV4i8nuO}\tag{I} y_t = \phi y_{t-1}+ u_t, \quad u_t \overset{iid}{\sim} N(0,\sigma^2) \end{equation} We ...
PSE's user avatar
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Weighted sum of two random variables ranked by first order stochastic dominance

Suppose $X$ and $Y$ are two non-negative, independent random variables such that $X \succcurlyeq_{st} Y$. That is, $X$ first-order stochastically dominates $Y$. Suppose that $X$ and $Y$ have smooth ...
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1 vote
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110 views

The effect of a small change of the probability distribution on the output of the function

Suppose $X$, $Y$, $X'$ and $Y'$ are random variables whose probability density follows the following relations. \begin{align} \|p_X-p_{X'}\|_{\mathrm{TV}}&\leq\epsilon_1,\\ \|p_Y-p_{Y'}\|_{\mathrm{...
Math_Y's user avatar
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1 answer
143 views

Positive definiteness and inverse of a symmetric matrix

Let $L_{\boldsymbol{\Delta}}\in \mathbb{R}^{n \times n}$ be the following symmetric matrix: $L_{\boldsymbol{\Delta}} = \boldsymbol{\Delta} - \frac{1}{\operatorname{tr}(\boldsymbol{\Delta})} \...
Mathieu's user avatar
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The expected value of a double exponential function of normal random variable

Let $X$ be a random variable from a normal distribution $N(\mu, \sigma)$. How do we calculate the expectation $E[e^{k\cdot e^{-X}}]$, where $k<0$? I think we can use the moment generating function ...
frostman's user avatar
4 votes
2 answers
112 views

Convex hull of bivariate normal random points

Let $n > 1$, and $p_1, \ldots, p_n \in \mathbb{R}^2$ be iid random points following a bivariate normal distribution (doesn't matter which one), and let $X_n$ be the number of vertices of convex ...
Mikhail Tikhomirov's user avatar
1 vote
2 answers
150 views

Realizing a negative-binomially distributed random variable simultaneously in two different ways

My actual question appears at the bottom of this posting. Suppose $$ \Pr(\Lambda\in d\ell) = \frac 1{\Gamma(\alpha)} \left( \frac\ell m\right)^{\alpha-1} e^{-\ell/m}\, \left( \frac{d\ell} m \right) \...
Michael Hardy's user avatar
1 vote
1 answer
113 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
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1 vote
1 answer
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Is the main part of certain exponential family sub-Gaussian?

$X$ is in the form of exponential family i.e. $$\mathbb{P_\theta}x = h(x)e^{\langle \theta,T(x)\rangle-\phi(\theta)}$$ where $\theta\in \mathbb{R}^d$. If $\nabla\phi(\theta)$ is L-Lipschitz i.e. $$\...
dc3506's user avatar
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2 votes
2 answers
136 views

An inequality for two iid r.v.s

Suppose $X$ and $Y$ are iid, show that $P(|X+Y|<1) \leq 3 P(|X-Y|<1)$.
Sheng Wang's user avatar
4 votes
1 answer
334 views

Inequality for Fourier transform of a power exponential function

Let $$ f_{\alpha}(x)=\phi_1(\alpha) \mathrm{e}^{-\frac{|x|^\alpha}{\phi_2(\alpha) }}, x \in \mathbb{R}, 0<\alpha<2, $$ where $\phi_1(\alpha)=\frac{\alpha}{2} \left\{{\{\Gamma(3/\alpha)\}^{1/...
Tanya Vladi's user avatar
5 votes
1 answer
339 views

Reference Request for a particular approach of (rigorous) statistical mechanics

I was reading Mathematical Aspects of Quantum Field Theory by. E. de Faria and W. de Melo, and the following caught my attention. In (Hamiltonian) mechanics, the states of a system are described by ...
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1 vote
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Urn model with delayed replacement

Suppose I have x red and y blue balls. At each timestep I draw a ball with probability $$P(\text{red ball}) = (x/(x+y))^z, P(\text{blue ball}) = 1-P(\text{red ball})$$ where z is fixed. Each ball is ...
timbuktu's user avatar
1 vote
1 answer
90 views

Bounding Kullback-Leibler

Suppose we have a probability distribution $P$ on a finite set $S$. We draw $N$ i.i.d. samples according to $P$ and use these samples to define an empirical distribution $R$. We measure the Kullback-...
Bill Bradley's user avatar
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2 votes
1 answer
116 views

A different way to try to define a measure on the unit-circumference circle

Let C denote the Lie group (ℝ/ℤ, +), and let ℤn denote the subgroup of C generated by [1/n]. Let (X,n) be an ordered pair where X ⊂ C is an arbitrary subset, and n ∊ ℤ+, such that that the set of ...
Daniel Asimov's user avatar
3 votes
0 answers
128 views

Direct analytic proof of positive definiteness of stable characteristic functions

Is there a direct analytic proof that the function $$ f ( t ) = \exp\left(-|t|^\alpha \big[ \lambda + i \theta \operatorname{sign} ( t ) \big]\right), \qquad \lambda > 0, \quad |\theta| < \...
tsnao's user avatar
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0 answers
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Alternating exponential distributions

Consider a process where objects arrive according to an exponential distribution with $rate=\lambda$. Let $X$ be the number that arrive over an interval of length $T$. Then the number that arrive is ...
ericf's user avatar
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1 vote
0 answers
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Distribution of norm over projected unit vectors

I am interested in the distribution of norms of projected unit vectors, for a particular class of projections. We first draw an $n$-dimensonal unit vector $v=X/||X||$ where $X=(X_1,X_2,\cdots, X_n)$ ...
galoistr93's user avatar
0 votes
1 answer
72 views

Analogues of Kac-Bernstein characterisation theorem for non-normal distributions

Let $X,Y$ be two independent random variables. The Kac-Bernstein theorem states that if $X+Y,X-Y$ are also independent, then $X,Y$ are Normal. Are there analogues of this theorem for non-normal, ...
TheSimpliFire's user avatar
1 vote
1 answer
67 views

Let $\alpha\in(0,1),d\in\mathbb N^+$ and $X,Y\in\mathbb S^d$ be uniform, what is $\Pr[\lVert X-Y\cdot\sqrt{1-\alpha} \rVert^2\le \alpha]$?

Suppose that $X,Y$ are independent random $d$-dimensional vectors each uniformly distributed on the unit sphere, and let $Z=Y\cdot\sqrt{1-\alpha}$ be a uniformly selected vector on a slightly smaller ...
J J's user avatar
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3 votes
1 answer
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Are there any known results on the probability distributions of perpetuities with power law discount rates?

Currently I am working on studying stochastic integrals of the form: $$Z_\infty = \int_0^\infty e^{-f(t)}\mathop{d}S_t$$ where $S_t$ is a Compound-Poisson process with Exponentially-distributed ...
jam jelly's user avatar
1 vote
0 answers
63 views

Expectation of moduli of roots

For a complex polynomial $\sum_0^n a_i z^i$ the sum of roots is readily given by the Vieta's formula. However, after trying to find out the sum of moduli of roots in terms of the coefficients, I feel ...
AgnostMystic's user avatar
2 votes
1 answer
263 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
2 votes
1 answer
48 views

p.d.f. of exponential family

I have a question about the p.d.f. of exponential family. Suppose $(X,\mathcal{F})$ is a measurable space and $\{F_{\theta},\theta\in \Theta\}$ is a distribution family on $(X,\mathcal{F})$. When $\{...
newbie's user avatar
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3 votes
2 answers
148 views

Getting Wasserstein closeness from a derivative estimate

In my setting, $\mu$ and $\nu$ are probability measures on $\mathbb{R}^{2}$ with compact support. For any function $f\in{C^{2}_{b}(\mathbb{R}^{2})}$, I have the estimate: $$ |\mathbb{E}_{\mu}(f)-\...
David Pechersky's user avatar
0 votes
1 answer
106 views

What is the probability space corresponding to the probability measure $\mathbb{P}_{p}$ in the context of this paper?

Here is the definition of the frog model we are interested in: "... consider the homogeneous tree $\mathbb{T}_{d}$, that is, the rooted tree in which each vertex has (is connected by edges to) $d ...
user1234's user avatar
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0 answers
26 views

Prove the explicit form of the ratio function in a Markov Chain

Let $(A_M^{\mathbb{Z}_+}, \Omega, P, \lambda)$ be a Markov Shift where $A$ is a finite alphabet set, $M$ is the admissibility matrix, $P = [P_{i, j}]_{i, j\in A}$ is a stochastic matrix that is ...
Sanae Kochiya's user avatar
0 votes
1 answer
98 views

How does this Bayesian updating work $z_i=f+a_i+\epsilon_i$

$z_i=f+a_i+\epsilon_i$ ,where $f\sim N(\bar{f},\sigma_{f}^2)$ ; $a_i\sim N(\bar{a_{i}},\sigma_{a}^2)$; $\epsilon_i\sim N(0,\sigma_{\epsilon}^2)$. We can see the signals $\{z_i\}$ where $i\subseteq {1,...
yunfan Yang's user avatar
3 votes
1 answer
371 views

Examples of convergence in distribution not implying convergence in moments

It is well know that the convergence in distributions does not necessarily imply convergence in expectation, but implies convergence in expectation of bounded continuous functions. Let $\{X_n\}$ be a ...
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