# 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 $...

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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 $\...

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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)...

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1
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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, \...

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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}...

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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 ...

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1
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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} \...

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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 ...

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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 $d\times d$ ...

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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)(...

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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 $x_i$ ...

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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 ...

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### 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{...

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0
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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] $...

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1
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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\...

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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 ...

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1
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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 ...

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### 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 ...

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votes

1
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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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### 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{...

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1
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### 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})} \...

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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
...

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### 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 ...

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### 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)
\...

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1
answer

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### 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 ...

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1
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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.
$$\...

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votes

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### An inequality for two iid r.v.s

Suppose $X$ and $Y$ are iid, show that $P(|X+Y|<1) \leq 3 P(|X-Y|<1)$.

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### 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/...

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### 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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0
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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 ...

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1
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### 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-...

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1
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### 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 ...

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### 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| < \...

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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 ...

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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 $X=(X_1,X_2,\cdots, X_n)$ ...

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1
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### 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, ...

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1
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### 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 ...

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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 ...

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0
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### 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 ...

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### 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 $\mathbb{P}(X_1=X_2=\ldots=X_n=0)$...

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### 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 $\{...

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### 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)-\...

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1
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### 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 ...

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### 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 ...

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1
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### 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,...

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### 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 ...