The probability-distributions tag has no wiki summary.

**3**

votes

**2**answers

352 views

### What is the maximum entropy distribution on the natural numbers?

On the reals $\mathbb{R}$, the maximum entropy distribution with a given mean and variance is the Gaussian distribution.
Let $\mu, \sigma > 0$. What is the maximum entropy distribution on the ...

**2**

votes

**0**answers

37 views

### existence of Markov operators not generated by transition probability function

Transition probability functions can always be used to generate Markov operators, correct? So is it correct to say that a Markov process is a collection of Markov operators? On the other hand, are ...

**0**

votes

**0**answers

47 views

### Estimating the mean of a bivariate distribution by randomly sampling variates and rounding them to integer coordinates

Imagine that we have a particle sampling positions on a two-dimensional plane according to a bivariate probability distribution: $A*e^{-(\frac{(x-x_0)}{2\sigma_x^2}+\frac{(y-y_0)}{2\sigma_y^2})}$, ...

**1**

vote

**0**answers

204 views

### Sampling a two-dimensional Gaussian distribution at points along an integer lattice

Please consider a two-dimensional Gaussian of the general form: $A*e^{-(\frac{(x-x_0)}{2\sigma_x^2}+\frac{(y-y_0)}{2\sigma_y^2})}$, where $C$ is the peak of the Gaussian, i.e. the point at which the ...

**3**

votes

**2**answers

290 views

### What's the probability of differences among n independent uniform distribution variables?

Given n independent random variables $x_1,x_2,...,x_n$, they have standard uniform distributions over [0,1]. Then what's the probability that there is at least one $|x_i-x_j| >= d$ for any ...

**2**

votes

**2**answers

150 views

### Unusual Differential Equation for CDF

Consider the following differential equation
$$F(cx) = F(x) + x F'(x)$$
for $c>1$.
Does this differential equation belong to a some well known class?
Is there a way to find all the solutions ...

**1**

vote

**0**answers

68 views

### Determining the position of a coordinate by binning Gaussian noise around that coordinate to lattice points with vertex-specific probabilities [closed]

(NOTE: I have changed and hopefully simplified this question by removing the section on randomly perturbing lattice points, and instead specifying that the counts at each vertex should be randomly ...

**2**

votes

**0**answers

71 views

### Multiple independent random number streams

This question is somehow related to this one.
Having multiple streams of pseudo-random numbers known to be independent and with a uniform distribution (U1, U2,...,Un) I want to do Monte Carlo ...

**1**

vote

**1**answer

484 views

### Distribution and moments of ratio of two beta variables?

If $X$ and $Y$ are two Beta random variables, I am interested in the distribution of their ratio $X/Y$. More specifically, I am interested in the moment generating function of this ratio. There is a ...

**2**

votes

**0**answers

148 views

### 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 ...

**1**

vote

**1**answer

177 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 ...

**0**

votes

**0**answers

76 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 ...

**0**

votes

**1**answer

96 views

### 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 $\mathbf{S}$ $=$ ...

**2**

votes

**1**answer

414 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 = ...

**2**

votes

**1**answer

504 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)

**0**

votes

**1**answer

156 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. $\mathcal{N}(0,1)$.

**4**

votes

**1**answer

228 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 ...

**2**

votes

**2**answers

128 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

**1**answer

358 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

**2**answers

280 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

**0**answers

312 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

**1**answer

34 views

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

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

**3**

votes

**2**answers

500 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

**0**answers

132 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

**1**answer

64 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

**0**answers

141 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

**1**answer

352 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

**2**answers

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

**2**answers

371 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

**2**answers

204 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

**1**answer

688 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

**2**answers

220 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

**1**answer

282 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

**0**answers

74 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

**0**answers

123 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

**1**answer

159 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

**0**answers

137 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

**1**answer

195 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

**1**answer

134 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

**1**answer

729 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

**1**answer

205 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

**1**answer

234 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

**1**answer

162 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

**2**answers

108 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

**1**answer

62 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

**4**answers

958 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

**1**answer

149 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

**1**answer

270 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

**1**answer

501 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

**1**answer

1k 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 ...