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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Why is it impossible to create a numerically balanced die with more than 120 sides?

I allow myself to contact you as a mathematics enthusiast. I have recently been intrigued by the concept of balance in dice and the assertion that it would be impossible to create a numerically ...
Matthieu Nauly's user avatar
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Resources to understand Lebesgue measure of Brownian motion's path [closed]

[https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Hansen.pdf][page 12] and [peter morters][page 47] Let $B$ be a stanrd Brownian Motion and $R$ a function defined on $\mathbb{R}^2$ such ...
sara's user avatar
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Uniqueness of the variance

The variance assigns a number to each of certain probability distributions on Borel subsets of $\mathbb R$. It has the properties of (1) shift-invariance, i.e. if $X$ is a random variable with a ...
Michael Hardy's user avatar
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Dirichlet series solution to Poisson Point Process question (repost from math.SE)

Reposting here because the bounty on the original math.SE post expired, with no solutions or comments received. For any discrete subset $S$ of $\mathbb{R}^d$, consider a digraph formed by placing an ...
Jim Ferry's user avatar
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Show that $\max_{P_X : X\in (0,1) } \left| \frac{\mathbb{E} [ f'(X) ]}{ \mathbb{E} [ f(X) ] } \right|$ is maximized by at most two mass points

Let $f$ be some given well-behaved function. Consider the following optimization problem overall probability distribution on $[0,1]$ \begin{align} \max_{P_X : X\in [0,1] } \left| \frac{\mathbb{E} [ ...
Boby's user avatar
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3 votes
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Relative entropy equality for a sequence of Bernoulli random variables

We are given two joint probability distributions, $p$ and $q$, of $n$ Bernoulli random variables $X_1, X_2, \ldots, X_n$. We denote by $p(x_k\mid x^{k-1})$ the probability $\mathbb{P}_p(X_k=x_k\mid ...
Penelope Benenati's user avatar
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Why MLEs are asymptotically efficient whereas method of moment estimators are not?

Under appropriate regularity conditions it is well-known that Maximum Likelihood Estimation (MLE) produces asymptotically efficient estimators in the sense that their asymptotic covariance is given by ...
Aaron Hendrickson's user avatar
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Equivalence of score function expressions in SDE-based generative modeling

I am studying the paper "Score-Based Generative Modeling through Stochastic Differential Equations" (arXiv:2011.13456) by Yang et al. The authors use the following loss function (Equation 7 ...
Po-Hung Yeh's user avatar
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A probability distribution, with Fourier transform smaller than $C \exp(-ct^2)$

Is there a probability distribution $\mu$ (with reasonably nice density $f$ on $\mathbb{R}$) such that the Fourier transform (aka. characteristic function) $\psi_\mu(t) = \int_{\mathbb{R}} e^{itx} \, ...
Tardis's user avatar
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Subgaussian norm of a symmetric $\{-1,0,1\}$ random variable

Let $p\in [0,1/2]$, and define $\xi$ as the symmetric random variable such that $$ \xi = \begin{cases} 1 & \text{ w.p. } p\\ 0 & \text{ w.p. } 1-2p\\ -1 & \text{ w.p. } p \end{cases} $$ so ...
Clement C.'s user avatar
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Expected value of MGIG distribution

I'm currently dealing with a Gibbs sampler of the multivariate generalized inverse Gaussian distribution (MGIG). In order to check the correctness of the sampler, I'd like to know the expected value ...
Stéphane Laurent's user avatar
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Distribution of covariance matrix of vectors of lognormal rvs

I know that the Wishart distribution is the distribution of the sample covariance matrix of a multivariate normal distribution. I am wondering if the analog distribution for a sample from a ...
qcd's user avatar
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Conditions for an ODE with convolution term to have a probability distribution solution

Suppose we have a simple ODE like: $$ y''(x) + 2ay'(x) + by(x) = 0 $$ with the condition $y(0)=0$ for $x\leq 0$. Then the solution on $(0,\infty)$ will be of the form $Axe^{-ax}$ when $a^2=b$, $Ae^{-...
user1598's user avatar
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Convergence of the row sums in a triangular null array with zero mean

Let $(X_{jn})_{1\leq j \leq n}$, $n\in \mathbb N$, be a triangular array of random vectors in $\mathbb R^d$ (the $X_{jn}$ are understood to be independent in $j$ for fixed $n$.). We say that the ...
PSE's user avatar
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An attempt to define expected value of a Riemannian manifold valued random variable - what'll go wrong?

Let $X:\Omega\to (M,g)$ be a random variable taking values in a Riemannian manifold $(M,g)$ with the Riemannian volume form denoted by $dvol_g(x).$ We know that there's no standard way to generalize ...
Learning math's user avatar
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2 answers
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How to analyze the value of convergence of functions of random matrices?

Consider a random i.i.d matrix $\mathbf{A}_{m\times n}$ with entries generated from a complex Gaussian distribution with zero mean and unit variance. I am interested in the large dimension analysis of ...
Math_Y's user avatar
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Spectrum of Moore-Penrose pseudo-inverse multiplied by a constant

Consider a random rectangular matrix $X\in\mathbb{R}^{N\times P}$ where each entry is drawn from iid distribution with mean $m$ and variance $s^2$, and denote $X^+$ the Moore-Penrose pseudo-inverse. ...
Uri Cohen's user avatar
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1 answer
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On a double sum involving binomial coefficients

For natural $n$, let \begin{equation} p_n:=2^{1-n}\sum_{v=1}^l \binom l{(v+l)/2}1(v\equiv l) \sum_{u=1-v}^{v-1}\binom k{(u+k)/2}1(u\equiv k), \tag{1}\label{1} \end{equation} where $k:=\lfloor(n+1)/...
Iosif Pinelis's user avatar
1 vote
2 answers
115 views

Sum and alternating sum of a series of Bernoullian variates

Consider the random variables $a_i,i=0,1,\ldots,n$ be random variables which take values from $\{-1,1\}$ independently and randomly with equal probability. Let \begin{align} S &= a_1+\cdots+a_n , ...
AgnostMystic's user avatar
1 vote
1 answer
101 views

What is this distributional closeness?

Let $P$ and $Q$ be two distributions over a sample space $\Omega$ which I would like to show are close under some choice of distance function. So far I have managed to show that there exists a subset $...
user43170's user avatar
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Does a subset with small cardinality represent the whole set?

Assume that we have heavy-tailed distribution $F(x)$ such that \begin{align} F(x)=\mathbb{P}[X\geq x]=x^{-0.5}. \end{align} Then, we produce $N$ independent samples $X_1,X_2,\ldots,X_N$ from this ...
Math_Y's user avatar
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2 answers
263 views

Simplification of hypergeometric Function

First of all I am not at all a math expert, but I have some working knowledge. That said, please excuse "dumb" questions. I am looking at the following process: Assume you are on the 2-...
WaveL's user avatar
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3 votes
2 answers
207 views

Recovering measure support from the sequence of I.I.D random variables

Assume we have a Borel probability measure $\mu$ in $\mathbb{R}^n$ and a sequence of $\mu$ distributed I.I.D. random variables $x_n$. Is there a limit formula for $supp(\mu)$, something like closure ...
Dmitri Scheglov's user avatar
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Can a "discrete" argument involving a gamma mixture of Poisson distributions be adapted to work for non-integers?

PREFACE Here I will begin with this preface that is nearly the same as what I wrote in an earlier question. Suppose $$ \Pr(\Lambda\in d\ell) = \frac 1{\Gamma(\alpha)} \left( \frac\ell m\right)^{\alpha-...
Michael Hardy's user avatar
4 votes
0 answers
219 views

Weight transfer proof of Turán’s theorem

Turán’s theorem, which states that a $K_{p+1}$-free graph contains at most $(1-1/p)\frac{N^2}{2}$ edges, can be proven in many different ways, as pointed out, for example in M. Aigner, G. M. Ziegler, ...
Martin Leshko's user avatar
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384 views

Bounding the variance of a truncated Gaussian random variable

Suppose $X_1, X_2, X_3 \sim N(0, 1)$ are three independent standard normal random variables. I am interested in showing that: $$\text{Var}[X_2\mid X_2 \geq X_1 - a, X_1 \leq X_3 + b] < 1,$$ where ...
B Merlot's user avatar
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1 answer
131 views

joint density of two relevant random variables

It seems that for most of the examples to derive the joint density of two or more random variables, the random variables themselves need to be independent. Is it possible to get the joint density of ...
Wang Jing's user avatar
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0 answers
87 views

The best probability distribution for the game of Number Master

In the game of Number Master, the player controls a number starting with $1$ and hits the other numbers one by one on the road. If the player hits a number smaller or equal to the current controlling ...
Y. Yang's user avatar
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3 votes
1 answer
135 views

Quantitative results (with formal proof) on the median approximation of Chi-squared distribution

It is well-known that Chi-squared distribution $X_n$ with degree-$n$ freedom has an approximate formula for its median as $n\left(1-\frac{2}{9n}\right)^3$. Or $(X_n/n)^{\frac{1}{3}}$ is approximately ...
taylor's user avatar
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2 votes
1 answer
124 views

von Mises Distribution property

The von Mises distribution has the highest entropy for a given first circular moment. It seems that the converse is true: for a fixed entropy, the von Mises distribution has the highest first ...
Jonathan Shaw's user avatar
4 votes
3 answers
376 views

Gaussian approximation of the characteristic function of Rademacher sum

I am searching an uniform bound for the characteristic function of some Rademacher sum. Specificaly I want to estimate how much the characteristic function is close to a Gaussian. We have in general ...
Nomaï's user avatar
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1 answer
156 views

Under what conditions is a multivariate normal distribution exchangeable?

Consider a multivariate normal distribution. What are necessary and sufficient conditions (esp. on the covariance matrix) under which this distribution is exchangeable?
user449277's user avatar
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How to find lower bounds of a modified mixing time (defined below) with respect to spectral of a finite Markov chain?

I am focused on a time-homogeneous continuous-time Markov chain with a finite state space $\mathcal{X}$, whose Markov kernel is $K$ and the corresponding semigroup is $H_t=e^{-t(I-K)}$. The invariant ...
Richard Ben's user avatar
1 vote
0 answers
52 views

A one-sided/monotone version of min/max-stable distributions -- does this have a name?

In a couple of papers I am working on (in random graph theory) I have encountered the following property of certain probability distributions, which I will describe shortly, and I am wondering if this ...
Joel Ottar's user avatar
2 votes
1 answer
107 views

prove with a probability of at least $1/e$: $\left\|\sum_{i=1}^k a_{i} P_{i}\right\|_2 \geq\left\|P_{1}\right\|_2$ holds

Let $a_i (i \in\{1...k\})$ be $k$ IID standard Gaussian random variables, $P_i$ are $d$-dimensional constant vectors. How to prove with a probability of at least $1/e$, $$ \left\|\sum_{i=1}^k a_{i} P_{...
Shuofeng Zhang's user avatar
3 votes
1 answer
165 views

Variance lower bound for natural exponential family

Let $Q$ be a probability measure on $\mathbb{R}$. Let $$Q_h(dy) = e^{y \cdot h} Q(dy) / M(h) \quad \text{where} \quad M(h) = \int e^{y \cdot h} Q(dy)$$ defined for $h \in (-c,\infty)$ with some $c &...
Oxonon's user avatar
  • 143
-2 votes
1 answer
130 views

Branching process with varying offspring distribution at each step

Consider a simple branching process $Z_0,Z_1,Z_2...$ such that at every discrete step, a particle splits into $k\geq1$ particles where $k$ follows a discrete distribution with probability mass $p(k)$. ...
stopro's user avatar
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0 answers
70 views

Do finite exchangeable random sequences behave asymptotically independently?

Let $(X_{n,k})_{k=1,\ldots,n}^{n\in\mathbb{N}}$ and $(Y_{n,k})_{k=1,\ldots,n}^{n\in\mathbb{N}}$ be triangular arrays of row-wise exchangeable random variables, that is for any $n$ and permutation $\...
Greg Zitelli's user avatar
4 votes
0 answers
118 views

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
  • 141
-1 votes
1 answer
68 views

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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2 votes
1 answer
112 views

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)...
D.D.'s user avatar
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0 votes
1 answer
85 views

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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2 votes
1 answer
159 views

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\...
Math_Y's user avatar
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5 votes
1 answer
886 views

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
2 votes
1 answer
248 views

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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0 answers
70 views

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
14 votes
1 answer
385 views

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
0 votes
0 answers
66 views

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
  • 21
1 vote
1 answer
181 views

$L_1$ 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
  • 23
4 votes
1 answer
417 views

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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