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2
votes
2answers
124 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 ...
2
votes
1answer
340 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 $\bar{x^2}$ ...
1
vote
2answers
265 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 ...
7
votes
0answers
235 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 ...
0
votes
1answer
31 views

Computing a particular expection for a family of distribution

Consider the family of distributions having the form $$f = ...
3
votes
2answers
405 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 ...
3
votes
0answers
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 $i,j$, ...
-1
votes
1answer
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.
5
votes
0answers
114 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
259 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
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 ...
7
votes
2answers
279 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
179 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 ...
4
votes
1answer
505 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
196 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
230 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
53 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
83 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
106 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, ...
0
votes
0answers
101 views

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 ...
1
vote
0answers
117 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
188 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
116 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 ...
10
votes
1answer
698 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
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 $n$ ...
4
votes
1answer
207 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
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) &= ...
0
votes
2answers
95 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
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 ...
13
votes
4answers
863 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
138 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
268 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
376 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 $X_i$ ...
2
votes
1answer
796 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
103 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} ...
1
vote
2answers
258 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
768 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
172 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 ...
1
vote
1answer
367 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
132 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
331 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
258 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
84 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
245 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
46 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
50 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
416 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
184 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
300 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
232 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 ...