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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### Bivariate Poisson-Binomial distribution

Suppose you have $100$ coins whose probabilities of obtaining the outcome "head" are $p_1,\ldots,\,p_{100}$. These probabilities are not necessarily equal each other. Consider the following random ...

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**3**answers

240 views

### Distribution of sum of two permutation matrices

Determinant and permanent of sum of two $n\times n$ permutation matrices can be arbitrarily different.
What is the distribution of determinant of sum and difference of two $n\times n$ permutation ...

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### Compare KS test and Wasserstein distance or Earth mover's distance

I have tried the following question in couple of exchange sites but I did not get any views or reply. I am asking here as I am kind of desperate for the answer, please be considerate. Any suggestion ...

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### Closed-form formula for Wasserstein distance between uniform discrete distribution and discrete distribution with same support

Let $x_1,\ldots,x_n$ be $n \ge 1$ distinct points in $\mathbb R^d$ and consider two discrete distributions on these points $\mu = (1/n)\sum_{i=1}^n\delta_{x_i}$, and $\nu = \sum_{i=1}^n\nu_i\delta_{...

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**1**answer

40 views

### Use Importance sampling for multimodal and multivariate distribution draws, how to choose proposal distribution?

I'm in trouble trying to generate samples following a particular distribution which is not numerically known perfectly. Let us consider a $R^n$ space provided with an orthonormal base $( e_{1},...,e_{...

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19 views

### Conditional distribution of maximum of the multivariate normal distribution

Suppose $X=[X_1,...,X_n]^T$ follows an $n$-dimensional multivariate normal distribution $\mathcal{N}_n(\mu_1,\Sigma_1)$ and $Y=[Y_1,...,Y_n]^T$ follows $\mathcal{N}_n(\mu_2,\Sigma_2)$, and $Y$ is ...

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427 views

### Entropy and total variation distance

Let $X$, $Y$ be discrete random variables taking values within the same set of $N$ elements. Let the total variation distance $|P-Q|$ (which is half the $L_1$ distance between the distributions of $P$ ...

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228 views

### The probability density function of the number of coins to first fill one box of $N$

Given $N$ boxes with the same capacity $C$, I toss coins into the boxes uniformly, one by one. When any one of the boxes is full, the sum of the coins in all boxes is denoted $S$. How to compute the ...

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**1**answer

179 views

### A modest generalization of the law of large numbers

Suppose I collect $2n$ independent samples of a probability density function $f$, which are separated into pairs $\{X_i^1, X_i^2\}$ for $1\leq i\leq n$. Suppose I now consider the set of all $2^n$ ...

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### Conditional PDF of a function of random variable [on hold]

Let $U_1 \sim N(0, 1)$, and $U_2 = |U_1|$, how to argue that the conditional PDF of $U_2$ is $f_{2|1}(V_2|V_1) = \delta(V_2 - |V_1|)$?

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51 views

### Lower bound on the sum of pmf squared of a hypergeometric distribution

I am working on a proof of correctness for an algorithm I came up with. I encountered the following problem en route. I would appreciate if anyone had some idea or could point me to the relevant ...

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59 views

### What is the order of the left tail of a mixture of non-central chi-square?

Let $\mu\sim N(0,1)$, $Z\sim N(\mu,1)$. Then $Z$ can be viewed as a mixture of Gaussians. It can also be viewed as a Gaussian but there is a prior for the mean.
Let $X\sim\exp(\lambda)$ where the ...

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191 views

### Updating Geman and Geman (1984) on image restoration

I am reading the seminal paper
Stuart Geman and Donald Geman, Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images, IEEE Transactions on Pattern Analysis and Machine ...

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### Supremum of expectations over a family distributions close to a base distribution

Let $p$ be a probability distribution on a space $X$ , $f:X \rightarrow \mathbb R_+$ be a measureable function, and $0 < \alpha \le 1 \le \beta < \infty$. Define a sub-collection $\mathcal Q \...

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60 views

### General distributions with the “transportation-cost inequality” property to piece log-concave distributions

It is now known [Otto et Villani 2000; Cordero et al 2006; etc.] that on an $n$-dimensional smooth Riemannian manifold $X$ and a probability measure $\mu$ on $X$ with density $d\mu \propto e^{-V}dvol$ ...

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142 views

### wasserstein distance between distributions with bounded ratio

Let $p$ and $q$ be probability distributions on a metric space $X=(X, d)$ with densities $dp$ and $dq$, such that there exists $0 < \alpha < \beta < \infty$ satisfying
$$
\alpha d p \le dq \...

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### 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....

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77 views

### Stochastic domination of Gaussian random vectors

Let $S$ be the class of all $2$ by $2$ matrices of the form
$$\begin{bmatrix}
1 & a \\
a & 1
\end{bmatrix},\, |a|\leq 1.$$
Is there a single matrix $M\in S$ such that for ...

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49 views

### Upper bound for $\sup_{W_d(Q, P) \le \epsilon} Q(A)$, where $W_d$ is the Wasserstein metric

Let $X=(X,d)$ be a metric space and let $W_d$ denote the Wasserstein metric induced by this metric, on the space of probability distributions on $X$. Let $\epsilon \ge 0$, $A$ be a Borel subset of $X$...

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### General conditions for Gaussian isoperimetric inequaliteis

Context
I'm doing some work in which i need to show that the blow-up $B_\epsilon$ of a Borel set $B$ has large measure for $\epsilon$ sufficiently large. I've found a very attractive tool for doing ...

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### Probability of exiting on the boundary for a monotone Lévy-type process

Let the continuous function $\ell:\mathbb R \times(0,\infty)\to[0,\infty)$ be a Lévy-type kernel, such that
$$
\sup_{x}\int_0^\infty \min\{1,y\}\ell( x, y)\,dy<\infty,
$$
and suppose that $\...

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**1**answer

78 views

### A distribution which is Wasserstein-close to a compactly supported distribution is almost compactly supported

I wonder whether this is true: If a distribution is very close (in the Wassertein sense) to another distribution with compact support, then the former must put only tiny amount of mass outside a ...

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### Approximate in $W_1$ sense, an empirical distribution with restriction of true distribution on a set

Let $\mu$ be a probability distribution on a metric space $X=(X,d)$ (to avoid unnecessary complications, assume the full filtration $2^X$) and let $x_1,x_2,\ldots,x_N$ be a sample of size $N$ drawn i....

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151 views

### What is the probability for a Binomial to be greater than other?

Let $X = B(n, p)$ and $Y = B(k, q)$ be two random variables with binomial distribution and, let $s$ be a positive integer. Assume that $n > k+s$ and $np \geq kq+s$.
What is the probability for $X$ ...

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### KL divergence and mixture of Gaussians

Do we have an exact formula to compute the KL divergence between 2 mixtures of Gaussians (i.e convex combinations of a finite number of Gaussian distributions)?
If not exactly known, are there good ...

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67 views

### Draw samples from distribitions in the neighborhood of a fixed distribution

Disclaimer
Sorry in advance for vagueness. I'm still trying to get my ideas right on this one.
Setup
So, let $P$ be a distribution on a Euclidean space $X$ with an $\ell_p$ metric, and let $P_\...

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### Definition and examples of operator-stable distributions

I was trying to understand the basic ideas of the operator-stable distributions. I found the papers by Hudson and Sato. However, unfortunately, I am being unable to understand the mathematical ...

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94 views

### How to estimate a total variation distance?

Let $X_1, \ldots, X_n$ be independent Bernoulli random variables. Then $Pr[X_i=1]=Pr[X_i=0]=1/2$. Let $X = (X_1, \ldots, X_n)$ and $v \in \{0,1\}^n$, $Y=v \cdot X$, $Z=Y-1$. Let
\begin{align}
\mu_1(x)...

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### Bounding a particular ratio of sums

Let $p_i = \frac{1}{w_i + x}$ for $w_i >0$, $1\leq i \leq N$, and $x \in [\min_i w_i^{2/3},\max_i w_i]$. We wish to bound the ratio of the expected value of $w/(w+x)$ under the probability ...

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89 views

### Reference Request: Simple Random Walk on $\mathbb Z$ is Unimodal

I am looking for a reference to the following claim. Let $X = (X_t)_{t\ge0}$ be a continuous time simple random walk. Then
$$
m \mapsto P(|X_t| = m) : \mathbb N \to [0,1]
$$
is (weakly) decreasing (or ...

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### Integral of product of Gaussian pdf and cdf [closed]

$\phi(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$ is the pdf of a standard normal distribution.
$\Phi(x) = \int_{- \infty}^x \phi(t) dt$ is the cdf of a standard normal distribution.
How does one ...

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### Reference request on numerical integration

Let $\rho:\mathbb R^d\to\mathbb R_+$ be a density function with finite first moment, i.e.
$$\int_{\mathbb R^d}~ \rho(x)dx~=~1 \quad \mbox{ and }\quad \int_{\mathbb R^d}~ |x|\rho(x)dx<+\infty.$$
...

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124 views

### Can the differential entropy of a continuous distribution with lebesgue integrable density be negative infinity?

Define the (differential) entropy for a density $f$ as
$$ H(f) :=-\int_{0}^{1} f(x) \log_{2}(f(x)) dx \, .$$
I am trying to find a $f \in L_{1}([0,1])$ such that $f\geq 0, \int_{0}^{1} f(x) dx = 1$...

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### Probability distribution of the Hadamard ratio of two degenerate multivariate Gaussian distributions?

This question pertains to the theory of Hadamard/elementwise functions of multivariate r.v.s/random vectors, which is unfortunately not a very popular topic:
References for the theory of Hadamard ...

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**1**answer

163 views

### A metric stronger than total variation

Let $P,Q$ be two distributions on a finite set $X$. Consider the following metric*
$$ d(P,Q) = \frac12\max_{\emptyset\neq A\subseteq X} ||P(\cdot|A)-Q(\cdot|A)||_1. $$
Obviously, the total variation ...

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### How to find the optimal convergence rate?

I have already asked that Question on Cross Validated:
Link
Suppose there is some data $X_{1},X_{2},\ldots,X_{n}$. We further suppose that there is some parameter $\theta$, for which we want to do ...

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### A Question on Distribution of Maximum of iid Random Variables

Let $1/2 < \alpha < 1$ and $k \leq \alpha/(1-\alpha) $ be given, and
$X_1, \ldots, X_k$ be iid random variables with distribution
$$\mathbb{P}(X_1 \leq x)
=
\frac{\sin(\alpha\pi)}{\pi} \...

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### Parametric distances on product spaces of measures

Disclaimer: Please excuse my loose language. I'm neither an expert in geometry nor probability. Please ask for clarification if something appears unclear or awkward to you.
Let $X$ be a topological ...

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### Pair of vectors multiplied by a random matrix and its inverse transpose are distributed randomly up to their dot product

Given arbitrary nonzero vectors $\vec{x}_1, \vec{y}_1, \vec{x}_2, \vec{y}_2 \in \mathbb{Z}^{n}_p$ ($p$ prime) with $\langle x_1, y_1 \rangle = \langle x_2, y_2 \rangle$, I am trying to show that: $(...

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### CLT for random sums: Anscombe's Theorem vs. “classical” version

Given a compound Poisson distribution
$$S(t):=\sum_{k=1}^{N(t)} X_{k}$$ with
$N(t)\in\mathbb{N},\,t\geq0$ a Poisson process with rate $\lambda.$
$X_{k}\in L^{2}$ are iid random variables, i.e. $\...

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### Is there any work on the distribution of the difference between samples and sample mean?

given $X_1,\cdots,X_n\overset{iid}\sim F$, where $F$ is a truncated normal, I wonder if there's something known about the distribution and specifically about the sub-Gaussianity of $X_i-\overline{X}$ ...

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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 ...

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**1**answer

186 views

### Minimizing the expectation of a functional of probability distribution subject to an entropy constraint

Consider a PDF $\pi(x)$ for $x\in[0,1]$, and the following functional
$$
F(\pi) = \mathbb{E}_\pi |x-y| $$
It is minimized by any point mass, so to avoid such degeneracy I'd like to lower-bound the ...

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**1**answer

190 views

### Density of a somewhat random set

The density of a set
$X\subseteq\omega$ refers to:
$\limsup\limits_{n\rightarrow\infty}\dfrac{C\cap n}{n}$.
Given a set of positive integers
$F= \{m_0<\cdots<m_{k-1}\}$,
let $C\subseteq \omega$...

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96 views

### Fastest convergence of sum of uniform independent distributions to a Gaussian

The sum of uniform i.i.d. random variables follows the Irwin-Hall distribution. Through observation it seems that the convergence is faster in comparison to the sum of uniform independent but not ...

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118 views

### Bound for a conditional expectation

Let $a_i, i=1, \ldots, n$ be real numbers. Let $\epsilon_i, \, i=1, \ldots, n$ be a random variables that take values $\pm 1$ with equal probability and $r_i, i=1, \ldots, n$ be random variables ...

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136 views

### Show that interval of maximum probability grows no faster than $\sqrt{n}$ for binomial distribution

Let $X \sim \text{Binom}(n, p)$ a binomial random variable. I want to show that : $$\forall 0 < t < 0.9, \quad \exists C, \quad \forall n >1, \quad \mathbb P\bigg(|X-np| \leq C\sqrt{n}\bigg) \...

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43 views

### Maximum Number of modes of $V=U+Z$ where $Z$ standard normal and $|U|\le a$

Let $f_V$ be a pdf of random variable $V$ where
\begin{align}
V=U+Z
\end{align}
and where $U$ and $Z$ are independent and $Z$ is Gaussian. Moreover, suppose that $|U| \le A$.
Can we find the upper ...

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**1**answer

199 views

### Closed form of :$\int_{-1}^1 x^{2k} (\operatorname{erf}(x))^k \,dx $ for $ k$ is even integer and :$\int _{0}^{t}\exp(-x^2 \operatorname{erf}(x))dx$

This question is related to my question here such that i want to find a closed form of $\int_{-1}^1 x^{2k} (\operatorname{erf}(x))^k \,dx $ , for $k$ is even integer because for odd integer is $0$ as ...

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**1**answer

194 views

### Reference request: discretisation of probability measures on $\mathbb R^d$

Given a probability measures $\mu$ on $\mathbb R^d$ with finite first movement, i.e.
$$\int_{\mathbb R^d}|x|\mu(dx)~~<~~+\infty.$$
My concern is to approximate $\mu$ some $\mu_n$ that is ...