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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3 answers
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Integral of product of gaussian CDF and PDF

Looking for an analytic solution to the integral below: $$ \int_{-\infty}^\infty \Phi\left(\frac{x - a}{\tau}\right) \phi\left(\frac{x - b}{\sigma}\right)dx $$ where $\Phi(\cdot)$ and $\phi(\cdot)$ ...
user489812's user avatar
4 votes
1 answer
195 views

Uniform iid sequence

The Rademacher functions are an explicit iid sequence with Bernoulli law. Does it exist an explicit construction of an iid sequence with uniform law?
Piero D'Ancona's user avatar
1 vote
1 answer
148 views

Joint pdf of N generally correlated (absolute values of) R.Vs as a joint pdf of (absolute value squared) random variables

The joint pdf of $|g_1|,|g_2|,\ldots,|g_N|$ is given by $P_{|g_1|,|g_2|,\ldots\ldots,|g_N|}(r_1,r_2,\ldots,r_N)=$ $$\prod_{ \substack{k=1 \\ \mu_1 \triangleq 0}}^N \frac{2r_k}{\sigma^2(1-\mu_k^2)} \...
aftab's user avatar
  • 11
0 votes
1 answer
205 views

Orthogonal polynomials w.r.t. an arbitrary measure

Consider a random scalar variable $X$ with arbitrary measure. I'm after a basis of polynomial functions $\{p_k\}_{k=0}^\infty$ which are orthonormal with respect to $X$ in the sense that \begin{...
dotdashdashdash's user avatar
12 votes
2 answers
520 views

Generating function for counting partitions with corners

A corner of an integer partition is a location at where a box can be added to its Ferrers diagram to give a new partition. E.g. the partition $\{1,1,1\}$ has two corners, and $\{1,2\}$ has three ...
Ryan Mickler's user avatar
1 vote
0 answers
75 views

Minima of a cdf of multivariate normal distribution with respect to a parameter

Let $\mathrm{X}\sim\mathcal{N}_{3}(\boldsymbol{\mu},\mathrm{\Sigma})$ where \begin{equation} \boldsymbol{\mu} = n[(\mu_1-\mu_2)\sqrt{\xi_1\xi_2/(\xi_1+\xi_2)}, (\mu_1-\mu_3)\sqrt{\xi_1\xi_3/(\xi_1+\...
SP SINGH's user avatar
3 votes
0 answers
278 views

Tail bound on trace norm / nuclear norm / Schatten-1 norm of Rademacher matrix

Let $0 < r \leq d$ integers. Let $X$, $Y$ be $d \times r$ matrices of independent Rademacher variables, that is, $X,Y \in \mathbb{R}^{d \times r}$ with entries $\pm1$ with probability $1/2$. I am ...
arriopolis's user avatar
3 votes
1 answer
260 views

Discrimination between set of binary distributions

Suppose we know two sets of distributions $A=\{p_1,p_2,\cdots,p_k\}$ and $B=\{q_1,q_2,\cdots,q_k\}$. We are given $C=\{r_1,r_2,\cdots,r_k\}$ such that $r_i=p_i$ for all $i$ or $r_i=q_i$ for all $i$. ...
gondolf's user avatar
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1 vote
0 answers
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If all moment of X are greater than all moment of Y, can we said something about their probability?

Consider a continuous probability distribution $f$ and two random variables $X, Y$ both are greater than equal to $0$ and they are not identical random variables. Let's say one can show that $E[X]^k \...
En-Jui Kuo's user avatar
1 vote
0 answers
51 views

What can we say about $\mathbb{E}[\mathrm{tr}A^{1/2}]$ for $A=\frac{1}{C}\sum_{i=1}^\infty c_i \alpha_i\alpha_i^\top \in\mathbb{R}^{m\times m}$?

Suppose we are given a summable sequence $(c_i)_{i\in\mathbb{N}}$ with $\sum_{i=1}^\infty c_i = C<\infty$ and independent $m$-dimensional, standard Gaussian vectors $\alpha_i\sim\mathcal{N}(0,I_m)$,...
Felix Kastner's user avatar
4 votes
1 answer
251 views

Bounds on discrepancy metric of product measures

Consider two measurable spaces $X_1 = (\mathbb{R}^m,\mathcal{B}(\mathbb{R}^m),\mu_1)$ and $X_2 = (\mathbb{R}^m,\mathcal{B}(\mathbb{R}^m),\mu_2)$ and the product spaces $$X_1^{q} = (\times_{i=1}^q\...
Ludwig's user avatar
  • 2,682
0 votes
0 answers
49 views

Inclusion-exclusion in a set of multivariables

I want to determine the risk of a multi-asset portfolio, in a way different from previous attempts. It is because current methods focus on just the correlation coefficient of two variables, while in a ...
R Salimi's user avatar
  • 221
1 vote
2 answers
208 views

Connection between invariant measure and positive recurrence for continuum state space markov chain

Let $\{ X_n(\omega,x)\}_{n \ge 0}$ be a Markov chain with and underlying probability space $(\Omega,\Sigma,\mathbb{P})$ and state space $X= \mathbb{S}^1$. Suppose this markov chain admits unique ...
Giuseppe Tenaglia's user avatar
9 votes
1 answer
451 views

What do category theorists know about "probabilistic metric spaces"?

I recently stumbled upon the notion of probabilistic metric space as a generalization of Lawvere's metric spaces, and I am very interested in understanding it deeper. In short, instead of a distance $...
fosco's user avatar
  • 13k
1 vote
1 answer
194 views

Bernoulli trials with small dependencies: asymptotics (central limit theorem, law of the iterated logarithm)

Let $\{X_k\}$ be a sequence of random variables, with $X_k\in\{+1, -1\}$ for $k>0$, generated as follows. First, define $S_n=X_1+\dots +X_n$, with $X_0=S_0=0$, and let $0<\beta<\frac{1}{2}$. ...
Vincent Granville's user avatar
2 votes
0 answers
119 views

Optimal Monte Carlo Trace Estimator

For a psd real symmetric $d\times d$ matrix $A$ and a function $f: \mathbb{R}^d \to \mathbb{R}$, with $f(x) := x^T A x$ we have that with $p(x) = \mathcal{N}(0_d, I_d)$ (i.e. standard multivariate ...
Sebastian Nowozin's user avatar
3 votes
1 answer
417 views

An inequality relating the Kullback-Leibler divergence of two discrete distributions with constant reference distribution

Suppose that $D_{KL}(p_1\parallel q)<1$ and $D_{KL}(p_2\parallel q)<1$. I'm trying to show that either $D_{KL}(p_1\parallel p_2)$ or $D_{KL}(p_2\parallel p_1)$ will have an upper bound close to ...
Harry Lorentz's user avatar
3 votes
2 answers
508 views

Is every discrete compound Poisson distribution a mixed Poisson distribution?

I asked and bountied this question at math SE but didn't get any answers, so I suspect that only experts (if anyone) may know the answer. The mixed Poisson distribution and compound Poisson ...
tparker's user avatar
  • 1,243
0 votes
1 answer
269 views

When can a convolution be written as a change of variables?

Suppose $X$ is a random variable with a density $f(x)$ such that $f(x)$ is a convolution of some density $g$ with some other density $q$: $$ f = g\ast q. $$ Under what conditions does $X=h(Y)$, where $...
edgar314's user avatar
3 votes
1 answer
439 views

$\cos(\frac{x}{3})\cos(\frac{x}{9})\cos(\frac{x}{27})\dots$ as $x \rightarrow \infty$

It is known that $$ \cos(\frac{x}{2})\cos(\frac{x}{4})\cos(\frac{x}{8})\dots = \frac{\sin x}{x} = O_{x \rightarrow \infty}(x^{-1}) $$ Is it true that $$ f(x) = \cos(\frac{x}{3})\cos(\frac{x}{9})\cos(\...
minhtoan's user avatar
  • 1,404
1 vote
1 answer
205 views

Using Hoeffding inequality for risk / loss function

I've got a question to the Hoeffding Inequality which states, that for data points $X_1, \dots, X_n \in X$, which are i.i.d. according to a probability measure $P$ on $X$, we find an upper bound for: $...
Mathematiger's user avatar
2 votes
1 answer
357 views

Self normalized sum of products of i.i.d. random variables

Let $p\in (0,1)$ and $X_1, X_2, ...X_n \sim \text{Bern}(p)$ be $n$ i.i.d. Bernoulli random variables, where the probability that $X_i$ is $1$ equals $p$. Fix $a,b>0$ different from $1$ that satisfy ...
James Farre's user avatar
1 vote
1 answer
116 views

Expectation value of random GUE matrix

Let $A$ be a matrix of the Gaussian unitary ensemble (GUE) and $v_1,v_2$ be two orthonormal vectors. I wonder if one can compute (or at least get a non-trivial lower bound on) the expectation value $$\...
Guido Li's user avatar
3 votes
0 answers
82 views

Is the Beta distribution decomposable?

Consider a random variable $Y$ that follows the Beta distribution with parameters $\alpha$ and $\beta$: $$ Y \sim {\rm Beta}(x | \alpha,\beta) $$ Is it possible to decompose $Y$ as $$ Y = X_1 + X_2 $$ ...
Daniel Turizo's user avatar
1 vote
0 answers
48 views

Characterizing set of IID average of symmetric positive semidefinite matrices matrices

Let $\mathcal{S}_+^d$ denote the family of real $d \times d$ symmetric (strictly) positive definite matrices. Define $\mathcal{P}_d$ to be those measures $\nu$ on $\mathcal{S}_+^d$ (assumed to have ...
Drew Brady's user avatar
1 vote
1 answer
80 views

Compressing covariance matrix from 3X3 to 2X2

thanks for answering my question. My question is, Let $v_3=[a,b,c]^T$ is a probabilistic 3-D vector variable which is distributed by zero mean Gaussian of dense covariance matrix $\Sigma_{3\times3}$. ...
user484806's user avatar
2 votes
0 answers
91 views

Approximating a probability density with a point set

Let $f$ be a "nice" probability density on $\mathbb{R}^2$, let $p=1/k$ for some fixed positive integer $k$, and let $\epsilon>0$. Are there any known statements of the following form? &...
Tom Solberg's user avatar
  • 3,929
1 vote
1 answer
90 views

Estimating the variance of Monte Carlo estimators for $F_Z$ and $f_Z$, $Z=X/Y$

This question has been migrated from the MSE. Background/Motivation: We have $Z=X/Y$ where $X$ and $Y$ are independent and $X\sim\mathcal N(\mu,\sigma^2)$. The density of $Y$ is not important here. ...
Aaron Hendrickson's user avatar
2 votes
1 answer
291 views

Law of large numbers for triangular arrays whose moments "look independent"

Let $(X_{n,k})_{k=1,\ldots,n}^{n\in\mathbb{N}}$ be a triangular array of random variables with finite moments of all orders, with no assumptions on their independence. Suppose that $$ \mathbb{E}\left[\...
Greg Zitelli's user avatar
3 votes
0 answers
107 views

$\epsilon$-net for the set of distributions

Let $S$ be a space of all distributions over a finite set $X$. What is the size of its $\epsilon$-net w.r.t. the total variation distance defined as $d(P,Q) = \frac12 \sum_{x \in X} |\Pr[P=x] - \Pr[Q=...
Artur Riazanov's user avatar
0 votes
0 answers
50 views

Using random matrix theory to calculate Lyapunov spectrum; relation between cumulative distribution and the spectrum

I am reading a paper using random matrix theory to calculate Lyapunov spectrum. What particularly confuses me is Why is the Lyapunov spectrum simply the inverse of $\chi(x)$ (the probabilistic ...
Charlie Chang's user avatar
0 votes
1 answer
340 views

Fokker-Planck: uniqueness and convergence to stationary distribution

Consider the Langevin equation ($N$-dimensional) with nonlinear drift term but expressible as a gradient of a function $U(\vec{x})$. Namely, consider the stochastic process described by the set of ...
user1172131's user avatar
1 vote
1 answer
230 views

Bounding $2$-Wasserstein distance and the $L^1$ distance

My questions come from the paper Logarithmic Sobolev inequalities for some nonlinear PDE’s written by F. Malrieu (May 2001) where author omitted a good amount of details to be filled. Suppose that $W$ ...
Fei Cao's user avatar
  • 700
1 vote
2 answers
119 views

Existence of a strictly proper scoring rule on a $\sigma$-algebra that is not countably generated

Given a $\sigma$-algebra $\scr F$ on $\Omega$, say that an accuracy scoring rule for $\scr F$ is a function $s$ from the set of all (countably additive) probabilities on $\scr F$ to the $\scr F$-...
Alexander Pruss's user avatar
7 votes
2 answers
3k views

Expected value of min of variables - what informations do I need?

I encountered a problem where I need to compute: $$\mathbb{E}(U) = \mathbb{E}(\min(X_1, .. , X_6))$$ The problem is that I have little information on the $X_i$. Basically I know $\mathbb{E}(X_i)$ and $...
Qise's user avatar
  • 247
1 vote
1 answer
94 views

Tight upper-bound on dependent events

Given $r$ r.v.s $x_k$: $x_k=1$ with probability $p_k$ and $0$ otherwise. Let $s_i=\sum_{j=1}^r c_{ij}p_j$ where $c_{ij}\in[0,1]$, $1\le i\le n$. Let $E_i$ denote the event $\sum_{j=1}^r c_{ij}x_j>(...
lchen's user avatar
  • 459
1 vote
0 answers
110 views

What are the Lévy processes with specific increments?

It is known that the increment of the Wiener process $W$ is drawn from a Gaussian distribution, i.e. $\Delta W \sim \mathcal{N}(0, \delta t)$. I wonder what are the Lévy processes with increments from ...
user482699's user avatar
3 votes
1 answer
93 views

On an integral of Gaussian CDFs

Let $c>0$ and $T>0$ be fixed. Denote by $F$ the Gaussian CDF, i.e. $F:\mathbb R\to\mathbb R$ is defined by $$F(x):=\int_{-\infty}^x \frac{1}{\sqrt{2\pi}} e^{-z^2/2}dz.$$ For every $a\in [0,1)$, ...
GJC20's user avatar
  • 1,220
-1 votes
2 answers
150 views

Cumulants of a sequence of variables with zero mean and variance

Can one prove for a sequence of positive random variable $X_{n}$ such that $\lim_{n\to \infty}E[x_{n}] = 0$ and $\lim_{n\to \infty}E[x_{n}x_{n}]= 0$ all the cumulants go to zero once $n\to \infty$ ?
gas's user avatar
  • 13
0 votes
1 answer
163 views

How to calculate the dual spaces of the following spaces?

Let us consider the spaces $C_\infty(E)$, $C_c(E)$, $C_b(E)$, where $E$ is locally compact, $C_\infty(E)$ is all continuous functions vanishing at the ends of $E$, $C_c(E)$ is all the continuous ...
Fractional analysics's user avatar
0 votes
1 answer
97 views

Does the the equivalence of Total variation distance formulas assumes that the two distributions are symmetrical?

Does the the equivalence of Total variation distance formulas presented here(https://ece.iisc.ac.in/~parimal/2019/statphy/lecture-14.pdf) assumes that the two distributions are symmetrical ?
Alup's user avatar
  • 11
1 vote
0 answers
53 views

A distribution $\pi \propto \exp(-f)$ satisfies log-Sobolev inequality, does $\exp(-af)$ also satisfy LSI?

Assume a distribution $\pi \propto e^{-f}$ satisfies log-Sobolev inequality (LSI) $$\forall \rho \in P(\mathbb{R}^n), \quad KL(\rho\| \pi) \le \frac{1}{2\lambda} I(\rho \| \pi)$$ with LSI constant $\...
JIaojiao Fan's user avatar
3 votes
3 answers
1k views

How close are two Gaussian random variables?

Given two Gaussian random variables A and B with (mean, standard deviation) of (a,s) and (b,m) respectively, is there a scalar w in [0,1] that indicates how close A and B are?
user1823664's user avatar
0 votes
0 answers
34 views

How to recalculate the weights for an event that happens multiple times and requires all outcumes to be unique?

I think it's easiest to explain with an example. I have a weighted probability list A : 0.15 B : 0.15 C : 0.15 D : 0.1 E : 0.1 F : 0.1 G : 0.1 H : 0.075 I : 0.075 ...
Darius Takacs's user avatar
0 votes
0 answers
65 views

Asymptotic distribution of a specific ML estimator

Consider a random independent sample of size $n$ from a distribution defined by the following probability density function$$f(x)=\frac{3}{4\theta}\cdot\left(1-\frac{x^2}{\theta ^2}\right)$$ when $-\...
Honza's user avatar
  • 419
1 vote
1 answer
211 views

The maximum trace of a covariance can be achieved by a discrete random vector?

Given a random variable $X$, satifying $P(0\leq X \leq 1)=1$, and $\mathsf{E}[X^2] = \alpha$. We know its maximum variance $\text{Var}(X) = \alpha(1-\alpha)$ achived by a binary random variable $P(X =...
Jone Sweden's user avatar
4 votes
1 answer
451 views

Transforming a Poisson distribution into a power law

Consider the probability mass function of the Poisson distribution given a mean $\lambda$: \begin{equation} \mathbb{P}\left(Y=k|\lambda\right)=\frac{e^{-\lambda} \lambda^{k}}{k !} \end{equation} By ...
stopro's user avatar
  • 67
0 votes
0 answers
62 views

The moment problem for $m_n=1/n$

What is the p.d.f. for the moments $m_n=1/n$ ? (They are obtained from $\int_0^1 x^n/x\ dx $, but clearly $1/x$ is not a p.d.f. on $[0,1]$)
Shadumu's user avatar
  • 85
5 votes
2 answers
138 views

Number of resampling until obtaining a uniform list

Let $A_0$ be a list of $ n$ distinct elements. By sampling with replacement the elements of $A_0$, we obtain a new list $A_1$ of $n$ elements that are not necessarily distinct. Repeat the same process ...
minhtoan's user avatar
  • 1,404
8 votes
0 answers
380 views

Non-affine smooth transformation of Gaussian is Gaussian

Suppose $Z\sim N(0,1)$ (standard Gaussian) and $f: \mathbb{R} \to \mathbb{R}$ is a differentiable function such that $f(Z)\sim N(0,1)$. My question is whether there exists any such $f$ other than $f(x)...
De vinci's user avatar
  • 329

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