Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
0answers
36 views

Upper bound on expectation of product

I want to upper-bound the following quantity: $$\mathbb{E}_Y\left[f(Y)g(Y)\right] $$ The idea would be to get something of the shape: $\mathbb{E}_Y[f(Y)]\cdot h(Y)$ where $h(Y)= j(\mathbb{E}_Y[k(g(Y))]...
0
votes
0answers
98 views

Which probability distribution has the most outliers?

Let $k$ be a positive real number. Which probability distribution over $\mathbb R$ maximizes $P(|x-E(x)|>k\cdot \operatorname{std}(x))$?
-1
votes
0answers
36 views

Understanding a probability formula of a random walk [on hold]

I have the following problem: Let's assume $G$ is a graph with vertices in red or blue colour. There is no limitation on how we connect the vertices, i.e., a red vertex can be connected with either ...
0
votes
2answers
90 views

Distribution of the individual coordinates of a uniform random vector on a high-dimensional sphere

Let $X=(X_1,\ldots,X_n)$ be a random vector uniformly distributed on the $n$-dimensional sphere of radius $R > 0$. Intuitively, i think that for large $p$ every coordinate $X_i$ is normally ...
1
vote
0answers
25 views

Choice of residual function for least squares error minimization

Good morning, I have the a set of data $(\sigma,D,\alpha_0)_i$, $i=1...n$ data. I want to determine two parameters $K_{IC}$, $C_f$ in the basic equation given as $K_{IC} = \sigma \sqrt{D} k_0(\...
0
votes
0answers
24 views

How can I solve a constrained optimization problem with a random number of decision variables?

Here is my problem. Let $A_t$ be a random variable with Poisson-Binomial distribution with set of success-probabilities $\{q_1^{(t)},\ldots,\,q_n^{(t)}\}$, with $t\in\{1,\,2,\,3,\ldots\}$, $n\in\...
2
votes
0answers
92 views

Random walk and comparing sums of Exponential random variables

Let $\sigma$ be the time a nearest neighbor random walk started at 1 that has probability $p>1/2$ of moving left reaches $0$. Let $\sigma'$ be an independent copy of $\sigma$. Let $(X_k)_1^\infty$ ...
1
vote
0answers
46 views

What are the moments of Kolmogorov Complexity for a Random Variable?

Given a random variable $X$ distributed under some computable distribution $P$ we have, $$0 \le E[K(X)] - H(P) \le K(P)$$ Where $H(P)$ is the entropy of $P$. I tried using Hoeffding concentration ...
0
votes
0answers
62 views

Asymptotic distribution of $n\mathbb E_{\hat{P}_n}[g(Z;\theta)]^T\operatorname{Cov}_{\hat{P}_n}[g(Z;\theta)]^{-1}\mathbb E_{\hat{P}_n}[g(Z;\theta)]$

Setup This question is a followup on this question. I'm interested in the asymptotic distribution of certain quadratic forms. So, let $Z$ be a $p$-dimensional random vector with (unknown) ...
1
vote
1answer
64 views

Generalization of inverse transform sampling

If X is a random variable over an arbitrary alphabet, is there a (deterministic) function f() such that X = f(U), where U is a uniform random variable over the unit-interval?
0
votes
1answer
24 views

Cumulative Order Statistics of Independent Non Identical Distributions

I understand that the p.d.f of order statistics for Independent Non Identical Distributions are given by the Bapat-Beg theorem as previously explained in another question. As explained in the article, ...
0
votes
0answers
25 views

Rate of convergence of centered Hotelling's statistic to Chi-squared distribution

Consider the Hotelling's statistic $H_n := n\mu_n\Sigma_n^{-1}\mu_n$, where $\mu_n$ (resp. $\Sigma_n$) is the empirical mean (resp. empirical covariance matrix) of a zero-mean random $d$-dimensional ...
0
votes
0answers
27 views

Distribution of an arithmetic progression with respect to a monotonic sequence with distributed increment

Let $T>0$, $\Delta>0$, $\Delta<T$, $\tau : N_e \rightarrow (0,\infty)$ be a sequence uniformly distributed on the interval $[T-\Delta,T+\Delta]$. Here $ N_e $ is the set of non-negative ...
3
votes
2answers
209 views

Effect of perturbing the atoms of a measure on the Wasserstein distance

Let $(X,d)$ be a metric space, $x_1,\ldots,x_N\in X$ and $x_1',\ldots,x_N'\in X$ be atoms, and $G=\sum_{i=1}^Np_i\delta_{x_i}$, $G'=\sum_{i=1}^Np_i'\delta_{x_i}$, and $G''=\sum_{i=1}^Np_i'\delta_{x_i'}...
-1
votes
0answers
20 views

A simple function that preserves the ordering of Binomial CDF

I am looking for a simple function $f(n, p, q)$ that exactly preserves the ordering of the cumulative probabilities $\mathbb{P}(\rm{Binom}(n, p) > n q )$, for all $n > 0$, $p, q \in [0, 1]$, ...
1
vote
1answer
136 views

continuity entropy with respect gibbs measures

Let $X=\{0,1\}^{\mathbb{N}}$. For simplicity I consider Bernoulli measures on $X$ only. Let $f:X\to \mathbb{R}$ be Holder continuous. The measure $\mu$ is a Gibbs measure with potential $f$ if there ...
0
votes
0answers
32 views

Expression for the Markov Chain CLT variance for an arbitrary initial distribution

Let $(\Omega,\mathcal A,\operatorname P)$ and $(E,\mathcal E,\pi)$ be probability spaces $(X_n)_{n\in\mathbb N}$ be a $(E,\mathcal E)$-valued stochastic process on $(\Omega,\mathcal A,\operatorname P)...
1
vote
0answers
25 views

Expected value of inverse of complex non-central Wishart matrix

I have a matrix $W$ that abides a complex non-central Wishart distribution. My question is what the expectation of the inverse is, i.e., how to compute $$\mathbb{E}(W^{-1}).$$ I have tried to read up ...
1
vote
1answer
132 views

Asymptotic distribution of $\mathbb E_{\hat{P}_n}[Z]^T\operatorname{Cov}_{\hat{P}_n}[Z]^{-1}\mathbb E_{\hat{P}_n}[Z]$

Under very general conditions on the random $p$-dimensional vector $Z$, what can be said about the asymptotic distribution of the (random) scalar quantity $R_n := \mathbb E_{\hat{P}_n}[Z]^T\...
1
vote
0answers
47 views

Families of distributions with a certain symmetry property?

Consider the probability distribution $\mathcal{N}_n$ on $\mathbb{R}^n$ whose density is $$(2\pi)^{-n/2}e^{-\frac{1}{2}||\vec{x}||^2} = \prod_{i=1}^n \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x_i^2}$$ ...
6
votes
4answers
257 views

Improvement of Chernoff bound in Binomial case

We know from Chernoff bound $P\bigg(X \leq (\frac{1}{2}-\epsilon)N\bigg)\leq e^{-2\epsilon^2 N}$ where $X$ follows Binomial($N, \frac{1}{2}$). If I take $N=1000, \epsilon=0.01$, the upper bound is ...
3
votes
3answers
172 views

What is the probability distribution of the $k$th largest coordinate chosen over a simplex?

Suppose we're selecting points uniformly at random from the $N$-simplex $S_N = \{x \in \mathbb R^{N+1}: $ all $ x_i \ge 0$ and $x_1 + \ldots x_N = 1\}$. One way to do this in practice is choose $N-...
1
vote
2answers
108 views

lower bound the probability of at least L collisions

Lets say we get a list $M$ containing $|M|=\sqrt{L\cdot N}$ randomly and independtly drawn elements from a set of size $N$. And lets denote the $i$-th element of the list $M$ by $M[i]$. If we now ask ...
2
votes
1answer
69 views

Controlling Mean Difference Between Product and Joint Distributions Using Optimal Transportation

Suppose we have nonindependent random variables $X \sim P$ and $Y \sim Q$, where $P$ and $Q$ denote their marginal distributions. We are interested in upper bounding $$ |\mathbf{E}_{X, Y\sim P \...
1
vote
1answer
33 views

Asymptotic eigenvalue distribution of sum of two i.i.d random matrices with Marchenko Pastur distributed eigenvalues?

Is there a method using random matrix theory and NOT using free probability to determine the asymptotic eigenvalue distribution of the random matrix $\mathbf{M}=\mathbf{X}_1+ \mathbf{X}_2$? where: $\...
3
votes
1answer
108 views

Log concavity of the maximum of dependent Gaussians

Let $Z_1,\dots,Z_n$ be dependent Gaussian random variables. Is it true that $X=\max\{Z_1,\dots,Z_n\}$ has a log-concave distribution function? This is true for the independent case, but is it true in ...
0
votes
1answer
87 views

Is the normal product distribution sub-gaussian?

Consider the normal product distribution, which is the distribution of the product of two or more independent normal variables. Particulary, focus in the case where the multiplied normal variables are ...
3
votes
1answer
114 views

Tail probability of random projection

Suppose $v\in R^n$ is a constant unit vector. $P_l$ is a random projection matrix to an $l$ dimensional subspace of $R^n$ which is uniformly sampled from $G(l,R^n)$ which is the collection of all $l$-...
0
votes
1answer
56 views

Bivariate Poisson-Binomial distribution

Suppose you have $100$ coins whose probabilities of obtaining the outcome "head" are $p_1,\ldots,\,p_{100}$. These probabilities are not necessarily equal each other. Consider the following random ...
7
votes
3answers
261 views

Distribution of sum of two permutation matrices

Determinant and permanent of sum of two $n\times n$ permutation matrices can be arbitrarily different. What is the distribution of determinant of sum and difference of two $n\times n$ permutation ...
1
vote
0answers
42 views

Compare KS test and Wasserstein distance or Earth mover's distance

I have tried the following question in couple of exchange sites but I did not get any views or reply. I am asking here as I am kind of desperate for the answer, please be considerate. Any suggestion ...
1
vote
0answers
49 views

Closed-form formula for Wasserstein distance between uniform discrete distribution and discrete distribution with same support

Let $x_1,\ldots,x_n$ be $n \ge 1$ distinct points in $\mathbb R^d$ and consider two discrete distributions on these points $\mu = (1/n)\sum_{i=1}^n\delta_{x_i}$, and $\nu = \sum_{i=1}^n\nu_i\delta_{...
0
votes
2answers
54 views

Use Importance sampling for multimodal and multivariate distribution draws, how to choose proposal distribution?

I'm in trouble trying to generate samples following a particular distribution which is not numerically known perfectly. Let us consider a $R^n$ space provided with an orthonormal base $( e_{1},...,e_{...
0
votes
0answers
25 views

Conditional distribution of maximum of the multivariate normal distribution

Suppose $X=[X_1,...,X_n]^T$ follows an $n$-dimensional multivariate normal distribution $\mathcal{N}_n(\mu_1,\Sigma_1)$ and $Y=[Y_1,...,Y_n]^T$ follows $\mathcal{N}_n(\mu_2,\Sigma_2)$, and $Y$ is ...
10
votes
3answers
503 views

Entropy and total variation distance

Let $X$, $Y$ be discrete random variables taking values within the same set of $N$ elements. Let the total variation distance $|P-Q|$ (which is half the $L_1$ distance between the distributions of $P$ ...
4
votes
2answers
241 views

The probability density function of the number of coins to first fill one box of $N$

Given $N$ boxes with the same capacity $C$, I toss coins into the boxes uniformly, one by one. When any one of the boxes is full, the sum of the coins in all boxes is denoted $S$. How to compute the ...
2
votes
1answer
189 views

A modest generalization of the law of large numbers

Suppose I collect $2n$ independent samples of a probability density function $f$, which are separated into pairs $\{X_i^1, X_i^2\}$ for $1\leq i\leq n$. Suppose I now consider the set of all $2^n$ ...
3
votes
1answer
52 views

Lower bound on the sum of pmf squared of a hypergeometric distribution

I am working on a proof of correctness for an algorithm I came up with. I encountered the following problem en route. I would appreciate if anyone had some idea or could point me to the relevant ...
1
vote
1answer
80 views

What is the order of the left tail of a mixture of non-central chi-square?

Let $\mu\sim N(0,1)$, $Z\sim N(\mu,1)$. Then $Z$ can be viewed as a mixture of Gaussians. It can also be viewed as a Gaussian but there is a prior for the mean. Let $X\sim\exp(\lambda)$ where the ...
3
votes
1answer
202 views

Updating Geman and Geman (1984) on image restoration

I am reading the seminal paper Stuart Geman and Donald Geman, Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images, IEEE Transactions on Pattern Analysis and Machine ...
1
vote
0answers
25 views

Supremum of expectations over a family distributions close to a base distribution

Let $p$ be a probability distribution on a space $X$ , $f:X \rightarrow \mathbb R_+$ be a measureable function, and $0 < \alpha \le 1 \le \beta < \infty$. Define a sub-collection $\mathcal Q \...
4
votes
1answer
131 views

General distributions with the “transportation-cost inequality” property to piece log-concave distributions

It is now known [Otto et Villani 2000; Cordero et al 2006; etc.] that on an $n$-dimensional smooth Riemannian manifold $X$ and a probability measure $\mu$ on $X$ with density $d\mu \propto e^{-V}dvol$ ...
3
votes
1answer
186 views

wasserstein distance between distributions with bounded ratio

Let $p$ and $q$ be probability distributions on a metric space $X=(X, d)$ with densities $dp$ and $dq$, such that there exists $0 < \alpha < \beta < \infty$ satisfying $$ \alpha d p \le dq \...
1
vote
0answers
27 views

Singular values of random matrices with inhomogeneous variances

If $X$ is a random rectangular matrix with independent identically distributed entries of zero mean and equal variance, then as $X$ gets big its singular values tend to a Marchenko-Pastur distribution....
2
votes
1answer
82 views

Stochastic domination of Gaussian random vectors

Let $S$ be the class of all $2$ by $2$ matrices of the form $$\begin{bmatrix} 1 & a \\ a & 1 \end{bmatrix},\, |a|\leq 1.$$ Is there a single matrix $M\in S$ such that for ...
1
vote
0answers
56 views

Upper bound for $\sup_{W_d(Q, P) \le \epsilon} Q(A)$, where $W_d$ is the Wasserstein metric

Let $X=(X,d)$ be a metric space and let $W_d$ denote the Wasserstein metric induced by this metric, on the space of probability distributions on $X$. Let $\epsilon \ge 0$, $A$ be a Borel subset of $X$...
0
votes
0answers
31 views

General conditions for Gaussian isoperimetric inequaliteis

Context I'm doing some work in which i need to show that the blow-up $B_\epsilon$ of a Borel set $B$ has large measure for $\epsilon$ sufficiently large. I've found a very attractive tool for doing ...
1
vote
0answers
29 views

Probability of exiting on the boundary for a monotone Lévy-type process

Let the continuous function $\ell:\mathbb R \times(0,\infty)\to[0,\infty)$ be a Lévy-type kernel, such that $$ \sup_{x}\int_0^\infty \min\{1,y\}\ell( x, y)\,dy<\infty, $$ and suppose that $\...
1
vote
1answer
79 views

A distribution which is Wasserstein-close to a compactly supported distribution is almost compactly supported

I wonder whether this is true: If a distribution is very close (in the Wassertein sense) to another distribution with compact support, then the former must put only tiny amount of mass outside a ...
0
votes
0answers
22 views

Approximate in $W_1$ sense, an empirical distribution with restriction of true distribution on a set

Let $\mu$ be a probability distribution on a metric space $X=(X,d)$ (to avoid unnecessary complications, assume the full filtration $2^X$) and let $x_1,x_2,\ldots,x_N$ be a sample of size $N$ drawn i....