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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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136 views

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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0answers
241 views

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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1answer
69 views

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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1answer
116 views

Linking error probability based on total variation

Consider probability measure $\mu_{XY}$ defined on $\mathbb{R}^d \times \{1,2,3\}$, and sub-probability measures $\mu_1$, $\mu_2$, and $\mu_3$ as $\mu_1(A):=P(X\in A, Y=0)$ and $\mu_2(A):=P(X\in A, Y=...
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1answer
63 views

How to simulate the fractional noncentral Wishart distribution?

I already asked this question on math.stackexchange but got no answer. For a non-integer number of degrees of freedom $\nu > p-1$, one can simulate the central Wishart distribution $W_p(\nu, \...
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2answers
192 views

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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2answers
257 views

Concentration inequality for sum of iid random variables that involve KL distance

Conider $X \in \mathbb{R}^d$ and $Y \in \{0,1\}$, and a joint distribution $p_{XY}(x,y)$, and a set of $N$ i.i.d. samples $\{(X_i,Y_i)\}_{i=1}^{N}$. Define $p_{X0} = p_{XY}(x,0)$ and $p_{X1} = p_{XY}(...
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349 views

How do I draw samples from this distribution?

Let S be the the standard K-1 simplex. Consider the following probability distribution: $$\begin{align} f(p,\alpha,\beta) &= \prod_{k=1}^K p_k^{\alpha_k-1}(1-p_k)^{\beta_k-1}\\ Z(\alpha,\beta) &...
3
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1answer
123 views

Product of estimates of mean values - Concentration of measure inequality

Let $X_{1},...,X_{d} \in \{-1,1\}^d$ be random variables, with $E[X_j]=\mu_j$. Having $n$ i.i.d. samples $x^{(i)}_1,x^{(i)}_2,....,x^{(i)}_d$, $i=1,...,n $, let $\hat{\mu}_{j}=\frac{1}{n}\sum^{n}_{i=1}...
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1answer
88 views

$\phi$ - Entropies - $\phi$ - Divergences and classical entropy recovery

The $\phi$ entropy is defined as $\text{Ent}_{\phi}[X]= \mathbb{E}[\phi (X)]-\phi(\mathbb{E}[ X])$ where $X$ is a random variabel and $\phi$ is a convex function ($\text{Ent}_{\phi}[X] \geq 0$). By ...
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0answers
222 views

What is the maximum entropy distribution over the integers

Let $μ=0,σ>0$. What is the maximum entropy distribution over the integers with mean $μ$ and variance $σ^2$? Is Skellam distribution a maximum entropy distribution? Is there a closed-form ...
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1answer
220 views

Wasserstein convergence of conditional measures

Suppose $W_r(\mu_n,\mu)\to0$, where $\mu_n$ and $\mu$ are discrete probability measures on some metric space $\Omega$, and that all measures have the same number of atoms $d$ (but not the same atoms): ...
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30 views

Matrix variate t-distribution and product of Beta distributions

This is a reference request for the following result. Let $X$ be a random matrix following the matrix variate $t$-distribution $T_{p,m}(\nu, M, U, V)$ (as defined in Wikipedia). Then $$ \frac{\det(U)}{...
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2answers
114 views

Minimum probability that two Gaussian random variables are small

Let $X,Y$ be two centered Gaussian random variables each with variance at most $1$. Note that we do not assume independence. I would like to minimize $$\mathbb{P}(|X|\leq 1, |Y|\leq 1).$$ Is it true ...
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0answers
53 views

Central Limit Like theorem for the distribution of F-statistics on all possible partitions?

I'd be happy for simply a reference or even search terms as I feel like this has to be known*. Suppose we have a known probability distribution $X$ and a fixed integer $n$. I am interested in the ...
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1answer
170 views

Tail bound of a distribution

Let $X_1, X_2, \ldots, X_n, Y_1, Y_2, \ldots, Y_n$ be independent binary random variables each being $1$ with probability $\frac{1}{k}$. Let $Z = X_1(Y_1 + \cdots + Y_k) + X_2(Y_2 + \cdots + Y_{k+1})...
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2answers
858 views

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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2answers
97 views

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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1answer
99 views

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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1answer
58 views

Maximizing the $\alpha$-moment of a distributution

Given $\alpha$ and constant $\mu$, $$\begin{array}{ll} \text{maximize} & \displaystyle\int_0^\infty p(x)x^\alpha \,\mathrm d x\\ \text{subject to} & \displaystyle\int_0^\infty p(x)\,\mathrm d ...
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0answers
87 views

Random projection increases the distance?

Consider two absolutely continuous random variables $X: \Omega \mapsto \mathbb{R}^d$ and $Y: \Omega \mapsto \mathbb{R}^d$ for probability spaces $(\Omega, \mathcal{F},p_X)$ and $(\Omega, \mathcal{F},...
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2answers
115 views

Estimate on lowest eigenvalue in GOE

I was wondering if there is an explicit estimate on the probability that the lowest eigenvalue of a $n \times n$ GOE matrix is larger than some number $x \in \mathbb{R}$. I am aware of the fact that ...
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0answers
52 views

Existence and uniqueness of fixed point in generalized condition of triangular norm

Definition 1) A Menger space is defined as a triple $\left( S,F,T \right)$ where $S$ is a set , $F$ is a collection of distribution functions and $T$ is a triangular norm function $T:[ 0,1 ]\...
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1answer
77 views

Where does the expected value in the restatement of the pseudoregret come from?

Given stochastic payoff functions $X_{1}(t) \dots X_{K}(t)$, each having a different probability distribution on $[0,1]$, denote the expected value of $X_i(t)$ by $\mu_i$, and define $\mu^* = \max_{i \...
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2answers
183 views

Closed-form solution for an integral involving the p.d.f. and c.d.f. of a $N(0,1)$-distributed random variable

Let $\phi(\cdot)$ and $\Phi(\cdot)$ be the probability and cumulative density functions, respectively, of a random variable with distribution $\text{N}(0,\,1)$. That is, $$\forall x\in\mathbb{R}:\,\...
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1answer
144 views

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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1answer
147 views

Integral involving Gamma function: density of Kendall-Ressel family of distributions

I came across the following function when reading the famous paper of Letac and Mora "Natural exponential families with cubic variance functions", i.e., $$f(x) = \frac{x^x e^{-x}}{\Gamma(x+2)}$$ for $...
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1answer
119 views

How to characterize Radon Nikodym's derivative of a coupling with respect to any measure in the product space?

In my math essay of thesis I have defined the probability coupling as follows $$\Pi(\mu,\nu)=\left\lbrace \pi \in \Omega \left\vert \begin{matrix} \pi(A\times\mathcal{Y})=\mu(A) \\ \pi(\mathcal{X} \...
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1answer
422 views

Is sum of dependent normal variables symmetric?

Consider two standard normal random variable, X and Y. They both have mean 0, and variance 1. But we don't know their dependency. Is it possible for X+2Y to be nonsymmetric? In another word, is it ...
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0answers
114 views

Anti-concentration bounds for folded normal and inverse of gaussian variables

Are there any easy to use bounds on sums of the following kind : $$ \sum_{i = 1}^{i = N} |a_i| \geq P \\ a_i \sim \mathcal{N}(0, 1) \\ $$ and also for sums of the form : $$ \sum_{i = 1}^{i = M} \...
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1answer
188 views

Probability of positive definiteness of a random matrix [duplicate]

Given an $n \times n$ symmetric random matrix whose entries have distribution $N(0,1)$, how to calculate the probability of positive definiteness of this matrix?
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57 views

Changing Couplings of Discrete Random Variables

Let $X,Y$ be two discrete random variables. Two joint mass distributions (couplings) with marginals $X$ and $Y$ and with entries $p_{i,j}=\mathbb{P}_1(X=i,Y=j)$ and $p_{i,j}'=\mathbb{P}_2({X=i,Y=j})$ ...
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1answer
132 views

Concentration inequality for maximum of gaussians

Let $Z_1,\ldots, Z_n$ be standardized Gaussian random variables and denote $\rho_{ij}=\mathbb{E}Z_iZ_j$. Can one give an asymptotically sharp bound for $$\mathbb{P}\,(\max_{1\leq i\leq n}Z_i>x), \...
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4answers
642 views

Is every probability measure a pushforward of Lebesgue measure?

If $m$ is a probability measure on a measurable space $(X, \Sigma)$, is there necessarily a measurable function $f : [0, 1] \to X$ such that $m(A) = \mu(f^{-1}(A))$ for all $A \in \Sigma$? ($\mu$ is ...
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0answers
45 views

Rate of $L_1$ loss in estmating density on $[0,1]$

Let $f$ be a density on $[0,1]$ and let $X_1,X_2,\ldots$ be $\textit{iid}$ $f$-distributed. Also, let $f_n$ denote the kernel density estimator, i.e. $$f_n(x) = \frac{1}{nh_n} \sum_{i=1}^n K\left(\...
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1answer
384 views

Inverse Laplace transform to get CDF

I have the following problem. If I can get some help, I would greatly appreciate it. I am trying to replicate a particular research paper and came across a problem: Suppose $X$ is a birth death ...
2
votes
1answer
119 views

Combining Couplings of Random Variables

Given a fixed positive integer $n$, I have two random variables $$A(n)=2^{A_2}\cdots p^{A_{p_n}}, B(n)=2^{B_2}\cdots p^{B_{p_n}},$$ where $p_n$ is the largest prime number not exceeding $n$, $(A_p)_{p\...
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1answer
158 views

On concentration of a sum random variable

Take a random variable defined as $$r=u_{11}v_{1}v_{1}+u_{12}v_{1}v_{2}+\dots+u_{n,n-1}v_{n}v_{n-1}+u_{nn}v_{n}v_{n}$$ where $v_{i}$ are independent uniform random variables from $\{0,\dots,b\}$, $u_{...
3
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1answer
89 views

A $t$-test for ordered pairs

Suppose I have random variables $$ W_i = \begin{cases} w_1 &\text{with prob. } p_1, \\ w_2 &\text{with prob. } p_2, \\ w &\text{with prob. } 1-p_1-p_2,\end{cases} \qquad i = 1, \dots, 2n+1....
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2answers
559 views

An inequality for expected value of normally distributed variables

Question. Let $X_1,\dots,X_n$ be random variables with normal distribution. Is it true that $$\mathbb E \prod_{i=1}^nX_i^{2k}\ge\prod_{i=1}^n\mathbb E X_i^{2k}$$for any $k\in\mathbb N$? (The ...
0
votes
1answer
127 views

For which $f$ does $\int f d\mu_n\to\int f d\mu$?

Let $\mu_n$ be a sequence of probability measures, and $\mu$ some other probability measure (all on the same space). Denote $C((\mu_n), \mu)$ the space of functions $f$ such that $\int f d\mu_n\to_{n\...
6
votes
1answer
108 views

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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1answer
73 views

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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1answer
54 views

Looking for a specific kind of a compactly supported one dimensional distribution

I am looking for a sequence of probability distributions (parameterized by $h \in \{1,2,3,4,..\}$) supported on the compact interval $x \sim [a(h),b(h)]$ such that, $a(h) > \frac{b(h)}{h^{\nu^2}} ...
6
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1answer
356 views

Probabilistic Proofs of Key Number-Theoretic Results

Given a positive integer $n$, let $p$ be the largest prime less than or equal to $n$. Let $N(n)=2^{C_2}\cdots p^{C_p}$ be uniformly distributed from $1$ to $n$, and $M(n)=2^{Z_2}\cdots p^{Z_p}$ where ...
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0answers
135 views

Random arithmetic formulas

An arithmetic formula is a well-formed expression involving only the constant 1, and the binary operations of addition and multiplication, with multiplication by 1 not allowed. For instance, $1 + (1 + ...
6
votes
3answers
326 views

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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0answers
95 views

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 ...
4
votes
2answers
59 views

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 ...
6
votes
3answers
221 views

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 ...