The probability-distributions tag has no wiki summary.

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### Quantifying the “flatness” of functions which are the Fourier transforms of positive functions

Short version of question: I'm trying to understand the extent to which a function is prevented from being "flat" as a result of being the Fourier transform of a positive function. That is, the extent ...

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65 views

### Distribution of Quadratic Forms in Rademacher Random Vector

This is a fairly well-known theorem on quadratic form in normal distribution, which is taken from here.
If $\mathbf y ∼ N(0, \sigma^2)$ is an $n\times 1$ column vector and $\mathbf M$ is an ...

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**1**answer

160 views

### Set of distributions that minimize KL divergence,

Assuming that $p,q$ are probability distributions defined on the same support $\{x_i\}_{0 \leq i \leq n}$, $\epsilon$ a small real number, and $D_{KL}$ the Kullback-Leibler divergence,
is there a ...

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69 views

### The supreme distribution of Brownian motion increment

Let $W_t$ be an one-dimensional standard Brownian motion, and $\theta_s$ is the shift such that $\theta_s( W_t)=W_{t+s}-W_s$, then are there any reference available regarding the distribution of the ...

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### Independence of Eigenvalues of Wishart

This question regards a previous post, but it is not immediately obvious the two are related, so I ask it anyways: are the eigenvalues of a Wishart matrix $\mathbf{S}$ $=$ ...

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**1**answer

325 views

### Expectation of Maximum of Uniform Multinomial Distribution

Suppose we have a uniform multinomial distribution with $k$ buckets, i.e. we put $n$ items uniformly at random in $k$ buckets leading to $n_1, \dots, n_k$ items in each bucket respectively. Let $m = ...

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**1**answer

301 views

### Order statistics of independent NOT identically distributed random variables

I want to find the p.d.f of the n-th order statistics from a set of independent, but NOT identically distributed random variables $X_1, \dots, X_n$ (the p.d.f. of the $X_i$'s is at hand)

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118 views

### concentration of sums of fourth moment of normals

I was wondering what is the best tail bound for
\begin{equation*}
\mathbb{P}\bigg\{\sum_{k=1}^n X_k^4>(1+t)3n\bigg\}\le ?
\end{equation*}
where $X_k$ are i.i.d. $\mathcal{N}(0,1)$.

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205 views

### Limit of pushforward measures of random variables is “represented” by a random variable

Suppose we have an arbitrary probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a sequence of real random variables $X_n:\Omega\to\mathbb{R}$ such that the pushforward measures ...

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125 views

### Drawing random variates from a partially described probability distribution

I have a probability distribution over $\{0,1\}^n$ but instead of knowing the full joint distribution $p(x_1,\dots,x_n)$, I only know $p(x_i=x_j)$ for each $i,j$. How could I draw a random binary ...

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342 views

### Central limit theorem for $P(x)\sim 1/x^3$ distribution

I have a random variable $x \in (0,\infty)$ with distribution $P(x)$ falling off slowly $P(x) \sim 1/x^3$ for large $x$. So the expectation value $\bar{x}$ is finite but the second moment $\bar{x^2}$ ...

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267 views

### What's the name of this distribution

What is the official name of this distribution,
$$k\exp(-\lambda|x|^\alpha),$$
where $\alpha$ is usually less than 1, but greater than 0? More importantly, could anyone tell me what the variance of ...

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245 views

### Inequality between incomplete beta and gamma functions; or when is binomial distribution function above/below its limiting Poisson

Please note: this question was posted first (September 4) in math.stackeschange.com and then (September 16) in stats.stackeschange.com. It got no answers in neither of those sites.
Let the ...

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**1**answer

31 views

### Computing a particular expection for a family of distribution

Consider the family of distributions having the form $$f = ...

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412 views

### Distribution of a product of two discrete i.i.d. variables

The problem is to estimate the distribution of product of two $\textit{discretized Gaussian}$ random variables with zero means. The discretized Gaussian means that the p.m.f. looks like
...

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115 views

### Concentration inequality for function of independent Bernoulli r.v.'s (related to random graph)

Consider a random undirected graph on a set of $n$ nodes, say $\{1,2,\ldots,n\}$, such that the probability of edge between nodes $i$ and $j$ is $p_{ij}$ (we may assume $p_{ij}=o(1)$ for all $i,j$, ...

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56 views

### Wishart random variables

I have a question about Wishart random variable. If X follows a Wishart distribution, then does X-Y follows a Wishart Distribution if Y is a Hermitian matrix?
Thanks.

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121 views

### Two sets of independent Bernoulli random variables

There are two sets of random variables $X_1,\ldots,X_n$ and $Y_1,\ldots,Y_n$ satisfying:
Each $X_i$ and each $Y_j$ has a symmetric Bernoulli distribution ($-1$ and $+1$ with probability $\frac12$ ...

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**1**answer

263 views

### Variance of euclidean norm of Gaussian vectors

Let $X$ be a Gaussian vector in dimension $n$, with $0$ mean and covariance identity. Let $A$ be a square matrix of size $n$, and $Y = A X$. Let $N$ be the square of $Y$ euclidean norm: $N = \sum ...

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112 views

### Distribution similar to PPP

According to the definition of Poisson Point Process, I can't define a certain number of nodes which are distributed in an area as PPP. Is there a distribution (a certain number of nodes distributed ...

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288 views

### How to efficiently sample uniformly from the set of p-partitions of an n-set?

Let $n,p \in \mathbb{N}_+$ with $p \leq n.$ Let $\mathcal{P}$ denote the set of partitions of $\{1, \ldots, n\}$ into $p$ nonempty sets. How can I efficiently sample uniformly from $\mathcal{P}$?

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182 views

### Uniform distribution in (non-compact) locally compact spaces

I'm trying to understand how much of the theory of uniformly distributed sequences in compact spaces can be extended to locally compact spaces.
Following L. Kuipers and H. Niederreiter - Uniform ...

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527 views

### What is the maximum-entropy distribution given mean, variance, skewness, and kurtosis?

$X\in \mathbb{R}$. Which distribution $P(X)$ has the highest possible entropy given its expected value, variance, skewness, and kurtosis? Is it an exponential family distribution of the form $P(X) ...

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### Asymptotic Expansion of Distribution in Central Limit Theorem for Non-Identically Distributed Random Variables

My question is related to the following theorem (e.g. Section XVI.4 of Feller's 1971 book): Let $Z_i$ $(i=1,\cdots,n)$ be independent and identically distributed random variables with mean zero, ...

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**1**answer

238 views

### Markov Chain: state reduction

Hi I am trying to understand a proof in a paper (written by Isaac Sonin), I don't know if anyone could give me a clarification on the following:
Firstly we have a Markov chain $\{Y_k\}$ with finite ...

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55 views

### Moments of Matrix Gamma distribution

Matrix gamma distribution (defined for example in http://en.wikipedia.org/wiki/Matrix_gamma_distribution) is one way to generalize Wishart distribution. In our course work that distribution was used ...

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86 views

### Sum of non-identical categorical random variables

Is there a named distribution for the sum of non-identical categorical random variables?
When the categorical variables are i.i.d., the sum is a multinomial distribution. When the categorical ...

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**1**answer

109 views

### What is a likelihood kernel?

The paper, "The Multinomial-Poisson Transformation" by S. Baker (see http://www.math.ntnu.no/inla/r-inla.org/papers/multinomial-poisson.pdf) presents "likelihood kernels" for multinomial variables, ...

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### proof of “supermodular function induces measure”

A function $f:\mathbb{R}^n\longrightarrow\mathbb{R}$ induces a measure by its finite differences, that is
\begin{align}
\mu((\mathbf{a},\mathbf{b}]) := \Delta_{a_1,b_1}\cdots\Delta_{a_n,b_n} f
...

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120 views

### Nontransitive dice: the least number of faces?

Here is an introduction to nontransitive dice. The question is: given $n$-player with a $m$-sided dice each one, the what is the minimum of $m$ for a fixed $n$ to produce nontransitivity?
Here is ...

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189 views

### star-product of copulas

I have recently come accross the star product of copulas, that is if $A$ and $B$ are 2-copulas and $\{C_t\}_{t\in[0,1]}$ is a family of copulas, then $C(x,y,z) = \int_0^y C_t(\frac{\partial}{\partial ...

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118 views

### Application of Toms- Stein restriction theorem for Strichartz estimates

The initial value problem for one dimensional Shrödinger equation is
$$iu_{t}+u_{xx}=0,$$
$$u(x, 0)= f(x),$$
where $u:\mathbb R \times \mathbb R \rightarrow \mathbb C$ is a complex valued ...

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702 views

### Does $P(X_1>X_2)$ and $P(X_1=X_2)$, where $X_1$ and $X_2$ are independent and Poisson distributed, uniquely determine the parameters?

Let $X_1$ and $X_2$ be independent Poisson distributed random variables with parameters $\lambda_1$ and $\lambda_2$, respectively.
Let $a = P(X_1 > X_2)$ and $b = P(X_1 = X_2)$.
Question: ...

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203 views

### First colour to be drawn $n$ times from a hypergeometric distribution

Consider a set up of drawing balls without replacement from a population of $R$ red balls and $B$ black balls. Is anyone aware of an efficient way to calculate the probability, $p$, that I draw $n$ ...

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215 views

### Measure concentration for weakly dependent random variables

For an application quite alien to probability theory, I'd like to have a kind of measure concentration estimate, in the following spirit. Suppose that to every $1\le i,j\le n$ there corresponds a ...

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152 views

### Distance between the product of marginal distributions and the joint distribution

Given a joint distribution $P(A,B,C)$, we can compute various marginal distributions. Now suppose:
\begin{align}
P1(A,B,C) &= P(A) P(B) P(C) \\
P2(A,B,C) &= P(A,B) P(C) \\
P3(A,B,C) &= ...

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97 views

### Joint distribution with specified marginals

Suppose we are given a probability distribution over a finite discrete product space $p(x,y)$ with marginals $p(x), p(y) > 0$ for each $x,y$ respectively. We are given two more marginal ...

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**1**answer

59 views

### Solutions of a stochastic reduced wave equation

Given a spherical reference frame $\left(\rho,\phi,\theta\right)$, the reduced wave equation can be written as:
$$\nabla^2U=k^2n^2U$$
in which:
$U=U(\rho,\phi,\theta)$
The solutions of this equation ...

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875 views

### A Normal Distribution Inequality

Let $n(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$, and $N(x) = \int_{-\infty}^x n(t)dt$. I have plotted the curves of the both sides of the following inequality. The graph shows that the following ...

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### Is anything known about Large Deviation Principle for non additive functionals on Markov chains?

Let $\Sigma$ be a finite set of cardinality $|\Sigma |$ and
$$\Pi = \{ \pi(i,j)\}_{i,j = 1}^{|\Sigma|}$$
a stochastic matrix (ie a matrix whose elements are non negative and such that
each row sum ...

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269 views

### Probability that no three events happen in a pre-defined window

Consider a Poisson process with arrival rate $\lambda$ arrivals per unit time. Given a window of time $W$ and a total of $k$ events, what is the upper bound of the probability that no three events ...

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388 views

### Generalized central limit theorem

I am looking for a generalized central limit theorem for non-square integrable stationary sequences. More precisely I suspect that when $(X_j)_{j\geqslant 1}$ is a stationary sequence such that $X_i$ ...

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### inner product of two gaussian random vectors?

Suppose that $x, y\sim N(0,I_n)$ are independent. Consider the inner product $\langle x, y\rangle$. Intuitively, $y$ behaves like a random vector of length $\sqrt n$, so $\langle x, y\rangle$ is close ...

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### Marginalizing multivariate normal over defined interval

Hello everyone,
I am trying to obtain an analytic expression for the following Gaussian integral
$$\frac{1}{\sqrt{(2 \pi)^n |\Sigma|}} \int \kern-0.2em \cdots \kern-0.2em \int d\mathbf{x}_{\sim i} ...

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264 views

### Probability distribution over cluster size in Erdős–Rényi random graph.

My question is about the probability distribution over the possible size of the containing cluster of a randomly chosen node in an Erdős–Rényi random graph.
Let G(n,p) be an Erdős–Rényi random graph ...

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777 views

### What is characteristic function of maximum of i.i.d. random variables?

Is is possible to get characteristic function of maximum of i.i.d. random variable sequence? Such as $X_1, X_2$ are two i.i.d random variables, then what is characteristic function of ...

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173 views

### Joint distribution from multiple marginals

Consider an experiment consisting of a repeated trial with two random Bernoulli (=binary) variables, A and B. Each trial consists of multiple outcomes for both A and B. Each trial has the same number ...

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382 views

### computing an integral involving standard normal pdf and cdf

recently, i need to compute this kind of integral:
$$ \int ^\infty _c \Phi(ax+b) \phi(x) dx$$
where a, b and c are all constants and $\Phi(x)$ denotes the CDF of standard normal distribution and ...

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133 views

### Sum of binomial probablilities

One of my friends is building a game where the player will get questions from 6 different categories. Each category has a total of 50 questions. A single game consists of answering one question from ...

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332 views

### Prove that the sum of a certain infinite series is 1

Prove the (numerically-evident) proposition that
\begin{equation}
\Sigma_{i=0}^\infty f(i) = 1,
\end{equation}
where
\begin{equation}
f(i)= 2^{-4 i-6} q(i) \frac{\Gamma(3 i+\frac{5}{2}) \Gamma(5 ...