The probability-distributions tag has no wiki summary.

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### 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 ...

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533 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 ...

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158 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 ...

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267 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
$\theta=tan^{-1}\frac{Y_1-Y_2}{X_1-X_2}+\pi$ ...

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294 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. ...

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**1**answer

145 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 ...

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233 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 ...

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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 $\lambda_{n,1},\dots,\lambda_{n,n}$ ...

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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 ...

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154 views

### Moments of the Kolmogorov distribution

Up to what order do the moments of the Kolmogorov distribution exist? References would be appreciated.

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316 views

### Expectation of $(c+e^{N(0,\sigma^2)})^{-n},\, n>0$

I would like to know if there's a way to compute or approximate the following expectation:
$$\mathbb{E}[(c+e^X)^{-n}]$$
where $X=N(0,\sigma^2)$ and $n,c>0$ (you can also assume that $n$ is a ...

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124 views

### Inequality for square of the subgaussian distributions

Hi all,
For my research I am trying to bound some exponential moments of subgaussian r.v.'s. And I am stuck with proving one of such inequalities. More specifically:
Let $a$ be unit vector in ...

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votes

**1**answer

164 views

### Central Limit Theorem for additive function of permutations of sequences

Fix the finite sets $\mathcal{X}$ and $\mathcal{Y}$, and probability mass functions $P_X(x)$ and $P_Y(y)$ on these sets. Assume each value of $P_X(x)$ and $P_Y(y)$ is rational.
For each $n$ such ...

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347 views

### Calculate channel capacity of general channel under constraint

Hi!
Given a conditional distribution $P_{Y|X}$ I'd like to find the prior distribution $P_X$ that maximizes the mutual information $I(X;Y)$ with $P_Y(y)=\int P_{Y|X}(y|x)P_X(x)\text{dx}$ (this ...

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**1**answer

129 views

### Statistical properties of principal components and their convergence rates.

Hello everyone, I'm interested in doing statistical tests on properties of principal components, but none of the literature I've found so far seems quite right for my purposes. Many articles present ...

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194 views

### Are all variables in a set of random variables independent if all pairs are independent?

If I have a sequence of random variables $X_1, X_2, \ldots, X_n$ (possibly infinite) such that all pairwise cdf's are factorized:
$$F(X_i, X_j) = F_i(X_i) F_j(X_j)$$
for all pairs $(X_i, X_j)$, does ...

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**1**answer

219 views

### Distribution of the powers of a primitive element of a finite field

What are known results regarding the distribution of the powers of a primitive element (generator of the multiplicative group) of a finite field? Specifically, compare the ordered list of ascending ...

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**1**answer

562 views

### Is an L_1 bounded sequence of random variables with uniformly converging CDFs uniformly integrable?

Changing my question in light of Dan's answer. Thanks, Dan.
Consider a sequence of real random variables $X_i$ bounded in $L_1$, that is $\mathbb E\left|X_i\right|\leq M$ for all $i$. Suppose that ...

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433 views

### Concentration bounds for sums of random variables of permutations

I'm trying to find theorems regarding random variables derived from sampling permutations, specifically concentration bounds.
As an example, let $X_i$ be the $\{0,1\}$-random variable that represents ...

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**1**answer

302 views

### Probability density that minimizes the sample range

Let $\mathcal{F}$ denote the set of all "concave probability distributions" on the unit interval, that is, all functions $f:[0,1]\to \mathbb{R}$ such that $f$ is concave, $f(x)\geq 0$ for all $x\in ...

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539 views

### What is the most extreme set 4 or 5 nontransitive n-sided dice?

A set of nontransitive dice is a set of dice whose face numbers are such that the relation "is more likely to roll a higher number than" is not transitive. (See wikipedia)
For some sets, the ...

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**1**answer

78 views

### Multinomial — how many trials in order to see all the values with prob 1-\alpha

Let suppose that I have a box with $k$ different balls, each one with a different color.
At each time I have to extract a ball and observe the color. Then I put the ball back in the box.
How many ...

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333 views

### Support of an infinitely divisible measure.

Hello,
if $G$ is a compact Lie group. Let $\mu$ be an infinitely divisible measure on $G$, such that $e$, the neutral element of $G$, is in the support of $\mu$. Is that true that the support of ...

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75 views

### Numerical optimisation for multivariate Gaussians

Hi,
I want to calculate
$
f_{\mathbf x}(x_1,\ldots,x_k)\, =
\frac{1}{(2\pi)^{k/2}|\boldsymbol\Sigma|^{1/2}}
\exp\left(-\frac{1}{2}({\mathbf x}-{\boldsymbol\mu})^T{\boldsymbol\Sigma}^{-1}({\mathbf ...

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656 views

### minimum of different independent Poisson random variables

Let $X_1,\ldots,X_N$ be independent Poisson distributed random variables with unequal parameters $\lambda_1,\ldots,\lambda_N$.
Is there any closed form expression or at least a good approximation for ...

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809 views

### Distribution of Inverse of a Random Matrix

Recently i got stuck into a problem and couldn't find its
satisfactory answer anywhere.
My question is simple. Suppose i have a fat random matrix (i,e $R$ has dimensions $k\times d$ where $k<d$) ...

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**1**answer

284 views

### Distribution function for divisors of an Integer

For a fixed $n$, let $D_n(x) = \{ d|n : d \leq x \}$ . We assume here $p \leq x \leq n/p$,
where $p$ is the smallest prime factor of $n$.
For example if $n = p^i$ for some prime $p$ then $D_n(x) ...

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### softmax activation function with infinite support ?

Hi,
How do we calculate the terms of a softmax activation function with an infinite support ?
That is, finding the $\{p_i\}_i$ with $p_i = {{e^{q_i}} \over {\sum_{j=1}^\infty e^{q_j}
}}$ (how to ...

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**1**answer

144 views

### First moment of a function of a normally distributed random variable

I'm trying to find the first moment of the following function:
$f(x) = \frac{(-ax+\sqrt{1-a^2})(-bx+\sqrt{1-b^2})}{\sqrt{x^2+1}}H(-ax+\sqrt{1-a^2})H(-bx+\sqrt{1-b^2})$ where $H(x)$ denotes the ...

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89 views

### Equivalence between choosing a subspace and choosing its orthogonal

Hi,
We consider subspaces of $\mathbb{R}^N$.
Suppose that we have a property called $\mbox{Prop}$ that apply to subspaces of $\mathbb{R}^N$. That is to say a function from the set of subspaces of ...

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292 views

### Simple markov chain problem

I know this is an easy problem, but I can't figure it out.
A particle takes discrete steps $σ_1,σ_2,σ_3,…,σ_n$ which take on values +1 or −1. However, $P(σ_i=+1)=p$ and $P(σ_i=−1)$ will be $1-p$.
...

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222 views

### Trasportation metric (AKA Earth-Mover's, Wasserstein, etc.) as “natural” / “induced”?

Context: Given a discrete finite metric space $X$ (in my case X={0,1}$^n$ with the Hamming/L$_1$ distance), I need to define the natural or canonical metric on the set of all probability distributions ...

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176 views

### Limit of the stochastic process at time 0

This is not a homework question so please be kind not to remove it right away. I am working on some research but have to justify the following argument: Assume $S_t$ is a continuous stochastic ...

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593 views

### Derivative of the most probable value (of a gaussian variable) VS most probable value of the derivative

Let $x$ be a random variable with gaussian probability distribution $P(x)$. We assume that $x$ depends parametrically on a parameter $t$ so that :
...

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98 views

### power law distribution of time events

Suppose you have the logs of a web server. In these logs you have tuples of this kind:
user1, timestamp1
user1, timestamp2
user1, timestamp3
user2, timestamp4
user1, timestamp5
...
These ...

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83 views

### Marginal Distribution.

Consider a couple of real random variables $(X,Y)$ and $\mu_{X,Y}$ the induced probability distribution.
Denote by $\mu_X$ and $\mu_Y$ the distributions of $X$ and $Y$.
Is it true that
...

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393 views

### Sampling without replacement until hitting a subset

I randomly sample uniformly from $ \{1,..,N \}$ without replacement until drawing a number $ \leq k$. Denote the expected number of draws by $R(N,k)$. I want a good approximation for $\sum_{k=1}^N ...

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**1**answer

178 views

### If two probability distributions have the same weak limit and one of them satisfies Large Deviation Principle, what can we say about the other?

If the probability distribution function of two sequences of random variables have the same weak limit and one of the sequences satisfies a Large deviation principle, then does it imply that the other ...

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194 views

### Sampling from a recursively defined distribution

I'd like to know if there are techniques for sampling from a recursively defined probability distribution, assuming that solving the recursion for a formula for the distribution is too difficult.
As ...

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278 views

### Product of probability densities of the form x^{-t} exp (-ax)

I have two probability distributions $p(x) = N_1 x^{-\tau} \exp(-\frac{x}{x_0})$ and $p(y) = N_2 y^{-\kappa} \exp(-\frac{y}{y_0})$. $N_1$ and $N_2$ are just normalization constants and $x>0$, ...

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164 views

### Expectation under a t-distribution

Given parameters $\lambda, \nu>0$, a covariance matrix $R$, a mean vector $\mu \in R^p$, the Arellano-Valle and Bolfarine's generalized $t$ distribution is given by (see, for example, the book by ...

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86 views

### Proving that a property holds for random sequences with given marginal distribution by rearrangement

I am currently investigating the property of random sequences with a special marginal distribution function $F(x)$. Given any random sequence $X_1, X_2, \cdots, X_n$, supposing their joint ...

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248 views

### Finding Decision Boundary from empirical distribution

Based on measuring a certain characteristic, we want to classify measurements as coming from either of two populations. The true population distributions are unknown (and we don't want to take any ...

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### Upper bound on expectation value of the product of two random variables [closed]

Hello,
I am trying to find an upper bound on the expectation value of the product of two random variables.
So suppose x, y are two non-independent random variables, given that I know the distribution ...

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119 views

### limit of functionals on weak convergent random variables

Suppose real value random variables satisfy
$X_{n} \Rightarrow X$ (convergence in distribution)
as $n\to \infty$ in the same probability space
$(\Omega, \mathcal F, \mathbb P)$.
It is well known that ...

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269 views

### Distribution of random vectors

Two positive numbers $\alpha$ and $\beta$ are given. We are going to describe a process of choosing a random vector on the unit sphere $S$ in $\mathbb R^3$ (given by $x^2+y^2+z^2=1$).
A vector $u\in ...

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**1**answer

129 views

### Ergodicity and convergence time in Probabilistic Cellular Automata

Has the following conjecture been prooved, or has any step in the direction of its proof been done?
"ANY Probabilistic Cellular Automata converge fast on the stationary probability distribution iff ...

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210 views

### Ergodicity for a Probabilistic Cellular Automaton on a finite space

Let's consider a Probabilistic Cellular Automaton on a one dimensional lattice $S$. Each site of the lattice can have two states, $0$ and $1$. The transition probability acting on each site is: ...

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2k views

### Cumulative distribution function and convolution.

Hello,
Given a probability distribution of a discrete variable p1(x) and a probability distribution of a discrete variable p2(y) defined by
p2(y) = Sum_{x,x'} p1(x) p1(x') * KroneckerDelta((x+x')/2 = ...

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394 views

### When can you describe a population and its component subpopulations with the same parametric family of distributions?

I believe that it is often the case that you are trying to select the best probability distribution to use to describe some phenomenon you are studying, and you have data not only for a population, ...