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1answer
68 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.
5
votes
0answers
147 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 $\frac12$ ...
-2
votes
1answer
411 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 ...
1
vote
2answers
114 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 ...
7
votes
2answers
401 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 $\mathcal{P}$?
3
votes
2answers
214 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 ...
5
votes
1answer
759 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) ...
1
vote
2answers
223 views

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, ...
0
votes
1answer
312 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 ...
1
vote
0answers
90 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 ...
1
vote
0answers
135 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 ...
1
vote
1answer
216 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, ...
1
vote
0answers
139 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 ...
2
votes
1answer
198 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 ...
0
votes
1answer
143 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 ...
11
votes
1answer
743 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: ...
2
votes
1answer
206 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 $n$ ...
4
votes
1answer
247 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 ...
1
vote
1answer
167 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) &= ...
0
votes
2answers
111 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 ...
2
votes
1answer
63 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 ...
14
votes
4answers
1k 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 ...
1
vote
1answer
150 views

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 ...
3
votes
1answer
276 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 ...
8
votes
1answer
546 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 $X_i$ ...
3
votes
1answer
2k views

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 ...
2
votes
0answers
149 views

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} ...
2
votes
2answers
335 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 ...
3
votes
1answer
960 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 ...
1
vote
0answers
227 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 ...
0
votes
1answer
674 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 ...
-1
votes
1answer
140 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 ...
1
vote
0answers
385 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 ...
-1
votes
1answer
338 views

expectation of ln(1+e^x) [closed]

x is a normal distributed variable. then what is the expectation of ln(1+e^x). i simulated this distribution and find that when x is N(0, 100), the mean of this function is around 4.1, and when x is ...
0
votes
0answers
99 views

Random variables related through nonlinear system of equations

I asked this question on http://math.stackexchange.com/questions/377140/random-variables-related-through-nonlinear-system-of-equations, however I received no answer for a while so I'm posting it here: ...
1
vote
2answers
251 views

A sampling and learning question

Suppose there is an oracle that returns a number $b \in \mathbb{Z}_{n}$ whenever I press the button. We have $b = a + e$, where $a \in \mathbb{Z}_n$ is a fixed number and $e$ is sampled according to ...
4
votes
0answers
48 views

Is there a name for the set of distributions whose probability generating functions are Mobius transformations?

Consider a discrete random variable $N\in\mathbb N$ with $\mathbb P(N=0) = p$, $\mathbb P(N=n) = (1-p)(1-q)q^n$ for $n\neq 0$. Then the probability generating function of $N$ $$\mathbb E(z^N) = ...
2
votes
0answers
67 views

Minimizing/Maximizing the tail of the convex combinations of Chi Squared i.i.d random variables

Consider $N$ i.i.d random variables, $X_{1}, X_{2}, \ldots, X_{N}$ , that are chi-squared of degree $K \geq 2$. Also consider the following 3 vectors: \begin{eqnarray*} \bar{a} &=& ...
3
votes
1answer
524 views

The average number of people that can sit on a bench of a given length.

Let me explain what I mean: The width of the average person varies, perhaps with a normal distribution. Given a specific variance, how many people (on average) can sit side-by-side on a bench of a ...
2
votes
1answer
226 views

Bounding statistical distance by matching moments

Suppose we have distributions $p(x)$ and $q(x)$ both supported on integers in $[-n, +n]$. We want $p$ and $q$ to have statistical (total variational) distance of at most $\epsilon$. Is there a ...
0
votes
1answer
391 views

Expected value with a kronecker product and Gaussian distributional assumption

What is the expected value, $ \mathbb{E}\left[ I \otimes \left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)\right]$ where $Z \sim N(0, \sigma^2I) $? The kronecker product is where the confusion ...
0
votes
0answers
335 views

Expected value of the log of the factorial of a poisson distribution

I found the expression for the expected value of the falling factorial of a Poisson distribution ($\lambda^n$) from - http://en.wikipedia.org/wiki/Factorial_moment. Is there a similar expression for ...
0
votes
2answers
559 views

Solution to the fractional differential equation

What is the solution of the fractional differential equation $$ f^{(\alpha-1)}(t) = tf(t) $$ where $(\alpha)$ denotes the fractional derivative of order $\alpha$ EDIT: Background behind this ...
2
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0answers
162 views

Have you seen this one parameter family of distributions before?

This is a one parameter family of distributions. Choose some parameter $\lambda > 0$ and define the measure $\nu_\lambda$ which is absolutly continuous with respect to the Lebsegue measure with the ...
0
votes
2answers
278 views

probability density function

Hey, what is the probability density function of the following random variable $\theta$ : $\theta=tan^{-1}\frac{Y_1-Y_2}{X_1-X_2}$ for $X_1-X_2>0$ and $\theta=tan^{-1}\frac{Y_1-Y_2}{X_1-X_2}+\pi$ ...
2
votes
2answers
348 views

Computing hypergeometric function of matrix argument

In the context of the Bingham probability distribution the ${ }_1F_1$ hypergeometric function of matrix argument naturally arises as a normalization constant of the probability distribution function. ...
1
vote
1answer
148 views

Weak convergence in measure for negligible sets.

Let $X$ be a Polish space and $(P_n)$ a sequence of Borel probabilities which converges weakly in measure to a Borel probability $P$. By this i mean that for any $f\in C_b(X)$ which is continuous and ...
5
votes
2answers
252 views

Liverani's CLT (a question)

Let $(\Omega,\mathcal{F},P)$ be a probability space where $\Omega$ is a complete separable metric space, let $T:\Omega\to \Omega$ ` be an ergodic transformation, let $\hat{T}:L^{2}_{_P}(\Omega)\to ...
2
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0answers
107 views

question about circular law

Hi, I have a question about the circular law. Consider $A_n=[x_{ij}]$ a sequence of random matrices where $x_{ij}$ are iid with mean $0$ and variance $1$. Consider $\lambda_{n,1},\dots,\lambda_{n,n}$ ...
0
votes
1answer
212 views

Order of difference of two generators of cyclic group

Let $n\in\mathbb{Z}^+$ and $\alpha,\beta $ be two generators of the cyclic group $\left(\frac{\mathbb{Z}}{(2^n - 1)\mathbb{Z}},+\right)$. Question: What are known theorems regarding the order of ...