The probability-distributions tag has no wiki summary.

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### Convergence rate of the central limit theorem near the center of the distribution

I'm looking for fast convergence rates for the central limit theorem - when we are not near the tails of the distribution.
Specifically, from the general convergence rates stated in the Berry–Esseen ...

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139 views

### explicit expressions of the distribution of sums of i.i.d. logistic random variables

Where can I find the explicit expression of the distribution of the sum of n i.i.d. logistic random variables, for n=2,3,4...
The expressions given in "On the convolution of logistic random ...

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185 views

### Expected value of swaps

Suppose you have a list of non negative numbers of size N. Now you calculate the maximum element in the list by scanning the list linearly and constantly updating a variable which has initial value of ...

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417 views

### Are there known expressions for total variation distance between $N(0,\sigma_1^2)$ and $N(0,\sigma^2)$

Are known expressions for total variation distance between $N(0,\sigma^2)$ and $N(0,\sigma^2+\epsilon)$ for small $\epsilon$? The only thing I seem to find is things are expression about the mean but ...

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81 views

### Fitting distribution to spatial data

I am studying a physical process generating data which projects nicely into two dimensions with non-negative values. Each process has a (projected) track of $x$-$y$ points -- see the image below.
...

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67 views

### Can truncated/non-smooth distributions be used as priors/posteriors in Variational Bayesian methods?

Variational Bayesian methods can sometimes be a good alternative to Markov Chain Monte Carlo numerical evaluation of probability distributions. They do this, as I understand it, by approximating the ...

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107 views

### approximation of probability distribution

I have a question: Let $\mu$ be a probability distribution defined on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ satisfying
$$\int_{\mathbb{R}}|x|d\mu<+\infty$$
Set
$$A_n=\Big\{\frac{i}{n}:~ ...

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624 views

### Euclidian norm of Gaussian vectors

Let $X \sim \mathcal{N}(0, \Sigma)$ be a Gaussian vector in dimension $N$. I am interested by the probability density of the random variable $\lVert X \lVert_2$.
If $\Sigma = {I}_N$, we recognize ...

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85 views

### Cramér-Wold like theorem for independent random variables

Let $X$ be a random vector in $\mathbb R^n$ with probability distribution $\mathbb P_X$. Now when given only the family of distributions
\begin{align*}
\left\{ \mathbb P_{v_1 X_1 + \dots + v_n ...

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118 views

### Does second order stochastical domination with increasing likelihood ratio imply first order domination?

This question is coming from the fact that all the counter examples for which second order stochastical domination holds but first oder stochastical domination fails do not accept increasing ...

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241 views

### Estimate on gaussian distribution

Let X be an $\mathbb R^d$-valued random variable with distribution $N_d(0,\Sigma)$. I'm looking for a function $f$ such that
$$P(|X_1|\leq M, |X_2|\leq M,\dots, |X_d|\leq M)\geq f(M),$$
and such that ...

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### Characteristic function known on subsets

Let $X$ be a random variable in $\mathbb R^n$ with distribution $\mathbb P_X$. Given a (infinite) family of matrices $W_t \in \mathbb R^{n \times m}$ parameterized by $t \in \mathbb R$, suppose we ...

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228 views

### Population dynamics for fish arriving via a Poisson process and living for a time given by some (not necessarily symmetric) general distribution

Imagine we have a hypothetical population of fish in a pond. The fish cannot reproduce, but are introduced by a Poisson process (with some known and fixed rate parameter independent of the total ...

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### Distribution of the square of a AR process

Suppose given a discrete time AR process $\{h_k\}$ with the following dynamics:
$h_k = (1 + a)h_{k-1}-b+v_{k-1}$,
where $a, b$ are constants and $v_k\sim \mathcal{CN}(0,1)$ is complex gaussian ...

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88 views

### Question about infinite-dimensional BM

Suppose we are given an $L^2(\mathcal{D})$-valued Brownian motion $W_t$ defined by
$$W_t:=\sum_{k=1}^{\infty}\sqrt{\sigma_k}W_t^k\phi_k(x),$$
where $\mathcal{D}$ is bounded domain in $\mathbb{R}^d$, ...

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### “Smallest” event such that probability greater than a given value

Very briefly, consider the probability space $(\mathbb R^n, \mathcal{B}(\mathbb R^n),P)$. During a problem I am studying, I came to a point where i need to compute
\begin{equation*}
\begin{aligned}
...

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237 views

### How do you call the problem of approximating a continuous distribution with a simple discrete distribution?

The following problem came up on the Mathematica forum as "Generating a list of integers that roughly satisfy a distribution": Given $n$, find $n$ integers (possibly with duplicates) whose ...

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48 views

### Is this exponential family Gaussian?

Let $m$ be a positive measure on the real line. Assume that $\exp k(t)=\int\exp(xt)m(dx)<\infty$ for $a<t<b$ and that $\exp k(t)=\infty$ if $t\notin[a,b],$, with $-\infty\leq a<b\leq ...

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138 views

### Computation complexity of calculating the cdf of an n-th dimensional gaussian random vector

Suppose you have a general $n$-th dimensional random Gaussian vector with probability distribution function $\mathcal{N}\left(\mathbf{x}|\boldsymbol{\mu},\boldsymbol{\Sigma}\right)$.
What is the ...

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### Maximal component of a multivariate Gaussian distribution

Suppose you have a general random Gaussian vector $\mathbf{X}\sim\mathcal{N}\left(\boldsymbol{\mu},\boldsymbol{\Sigma}\right)$. I'm looking for the simple way to calculate the distribution of the ...

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### Obtaining the 'threshold' of a distribution

Context of Research
Consider the expression:
\begin{align}
\widehat{\Theta}(\rho)_i = \frac{1}{(1-\rho)\Delta t} \ln\left(\frac{1}{T} \sum_{t=1}^T \left(\frac{1+r_t}{1+rf_t}\right)^{1-\rho} \right) ...

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### Eigen value distribution of autocorrelated Wishart matrix

Suppose the matrix W is constructed as $W=XX^T$ where $X_i(t) = \phi_i X_i(t-1) + a_i(t)$, and $a_i(t)$ ~ $N(0,1)$. I am interested in knowing the eigen value distribution of W. My google search on ...

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### Repeated draws from multinomial distribution

(This is a cross-post from Math StackExchange http://math.stackexchange.com/questions/609641/multinomial-distribution-sum-of-squared-probabilities)
Let $\vec X = (X_1, \dots, X_k)$ be a draw from a ...

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319 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 ...

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### 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 ...

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### 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: $A*e^{-(\frac{(x-x_0)}{2\sigma_x^2}+\frac{(y-y_0)}{2\sigma_y^2})}$, ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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

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380 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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451 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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149 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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### 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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### 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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### 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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### 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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### 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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### Computing a particular expection for a family of distribution

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

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### 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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### 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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### 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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### 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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### 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 ...