# 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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### 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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### 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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### 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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### 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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### 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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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/... 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 ... 1 vote 0 answers 68 views ### 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 ... 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-... 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 ... 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 $$f ( t ) = \exp\left(-|t|^\alpha \big[ \lambda + i \theta \operatorname{sign} ( t ) \big]\right), \qquad \lambda > 0, \quad |\theta| < \... 0 votes 0 answers 42 views ### Alternating exponential distributions Consider a process where objects arrive according to an exponential distribution with rate=\lambda. Let X be the number that arrive over an interval of length T. Then the number that arrive is ... 1 vote 0 answers 74 views ### Distribution of norm over projected unit vectors I am interested in the distribution of norms of projected unit vectors, for a particular class of projections. We first draw an n-dimensonal unit vector v=X/||X|| where X=(X_1,X_2,\cdots, X_n) ... 0 votes 1 answer 72 views ### Analogues of Kac-Bernstein characterisation theorem for non-normal distributions Let X,Y be two independent random variables. The Kac-Bernstein theorem states that if X+Y,X-Y are also independent, then X,Y are Normal. Are there analogues of this theorem for non-normal, ... 1 vote 1 answer 67 views ### Let \alpha\in(0,1),d\in\mathbb N^+ and X,Y\in\mathbb S^d be uniform, what is \Pr[\lVert X-Y\cdot\sqrt{1-\alpha} \rVert^2\le \alpha]? Suppose that X,Y are independent random d-dimensional vectors each uniformly distributed on the unit sphere, and let Z=Y\cdot\sqrt{1-\alpha} be a uniformly selected vector on a slightly smaller ... 3 votes 1 answer 105 views ### Are there any known results on the probability distributions of perpetuities with power law discount rates? Currently I am working on studying stochastic integrals of the form:$$Z_\infty = \int_0^\infty e^{-f(t)}\mathop{d}S_t$$where S_t is a Compound-Poisson process with Exponentially-distributed ... 1 vote 0 answers 63 views ### Expectation of moduli of roots For a complex polynomial \sum_0^n a_i z^i the sum of roots is readily given by the Vieta's formula. However, after trying to find out the sum of moduli of roots in terms of the coefficients, I feel ... 2 votes 1 answer 263 views ### On the mean value taken by Bernoulli random variables with joint distribution constraints We are given a vector n-dimensional random vector \mathbf{X} whose components are the Bernoulli random variables X_1, X_2, \ldots X_n, such that the probability \mathbb{P}(X_1=X_2=\ldots=X_n=0)... 2 votes 1 answer 48 views ### p.d.f. of exponential family I have a question about the p.d.f. of exponential family. Suppose (X,\mathcal{F}) is a measurable space and \{F_{\theta},\theta\in \Theta\} is a distribution family on (X,\mathcal{F}). When \{... 3 votes 2 answers 148 views ### Getting Wasserstein closeness from a derivative estimate In my setting, \mu and \nu are probability measures on \mathbb{R}^{2} with compact support. For any function f\in{C^{2}_{b}(\mathbb{R}^{2})}, I have the estimate:$$ |\mathbb{E}_{\mu}(f)-\... 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 ...
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 ...