# Questions tagged [probability-distributions]

In probability and statistics, a probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference.

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### A question about finite free convolution

For any square matrix $Y$ let $\chi_x(Y) = det(xI -Y)$ denote its characteristic polynomial. Say $A$ and $B$ are two $n-$dimensional symmetric matrices with constant row sums $a$ and $b$. Lets ...
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### Distribution of top singular vector of Bernoulli ensemble

Consider the distribution given by taking the top singular vector of a matrix whose entries are equally likely to be +1 or -1. This is not well-defined for matrices where the subspace achieving the ...
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### When a product of cdf and tail distribution is increasing?

My problem is the following. Given $F$ and $G$ cumulative distribution functions, with densities $f,g$ (for example on $[0,1]$), what can I say on the monotonicity of $F(x)(1-G(x))$? More specifically:...
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### Given the joint probability distributions of $X$ and $Y$ for $Y = R\,X+C$, find the probability distributions of $R$ and $C$

Let $R$, $C$, and $X$ be independent random variables defined on $(0,\infty)$ and $$Y=\underbrace{R\, X}_{Z}+C.$$ We are given the joint probability distribution of $X$ and $Y$, $P_{XY}(x,y)$ and ...
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### Lower bounds on Kullback-Leibler divergence

This was originally a question on Cross Validated. Are there any (nontrivial) lower bounds on the Kullback-Leibler divergence $KL(f\Vert g)$ between two measures / densities? Informally, I am ...
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### Ising model with zero external field - marginalization

The pmf of Ising model is considered as $p(\boldsymbol{x})=\frac{1}{Z(\theta)} exp\left\{ \underset{\left(s,t\right)\in E}{\sum\theta_{st}}x_{s}x_{t}\right\},\quad \boldsymbol{x}\in \{-1,1\}^n$, where ...
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### maximum likelihood estimation of X is better than that of f(X)?

Consider a binary variable $C$ with $\Pr(C=0)=\Pr(C=1)=0.5$. Consider a random vector $X \in \mathbb{R}^d$, characterized by $C$, with PDF $p_m(x)$, $m\in\{0,1\}$. Define a maximum likelihood (ML) ...
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### Does CLT hold for joint distribution of two dependent binomial variables?

Let $S_n$ and $T_m$ be two binomial variables satisfying $S_n\sim B(n,\frac12)$ and $T_m\sim B(m,\frac12)$. Define $\tilde{S}_n=\frac{2S_n-n}{\sqrt{n}}$ and define $\tilde{T}_m$ similarly. For any ...
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### Existence of distribution for certain order statistics

This is an open question: given a sequence of $n$ real numbers $x_1<x_2<\dots<x_n$, does there always exist a probability distribution, such that $\{x_i\}$ happens to be the $n$ expected ...
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### Is there a name for the sample variance process of a Lévy process?

Let $X_t$ be a Lévy process which is known to have mean zero and finite variance $t \cdot \sigma^2$, but for which the value of $\sigma^2$ is unknown. How do we estimate $\sigma^2$? One approach would ...
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### Central Limit Theorem when the sum of the variances is finite

Suppose $(x_i)_{i\in\mathbb{N}}$ a set of strictly positive numbers such that $L=\sum_{i\in\mathbb{N}}x_i$ is finite. Suppose that $(X_i)_{i\in\mathbb{N}}$ is a set independant (real-valued) random ...
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### log-like distance between probability distributions

Given two probability density functions (PDF) $f$ and $g$, both defined over the same set $X$, there are many ways to describe/measure the distance between them, e.g., KL divergence and Hellinger ...
### A density function that matches the $k$ smallest elements of $n$ uniform samples
Let me apologize in advance as this feels like a homework question, though I've tried without success to work through the relevant integrals. My question is: given positive integers $k$ and $n$ with ...
### Solve $\inf_{ X: |X| \le a \text{ a.s.}} E \left[ \frac{1}{1+(X-X^\prime)^2} \right]$
I want to solve the following optimization problem \begin{align} \inf_{ X: |X| \le a \text{ a.s.}} E \left[ \frac{1}{1+(X-X^\prime)^2} \right] \end{align} where $X^\prime$ is an independent copy ...