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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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122 views

What's the probability of two independent events in time domain?

Suppose there are two independent events A and B. The probability that A or ...
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0answers
62 views

Probability distribution from standard domain (two primes) - IV

Pick a random pair $(a,b)\in\mathbb Z_n^2\setminus\{0,0\}$. Denote $N_2(a,b,n)$ to be minimum $\ell_2$ norm of vector $(x,y)$ as $(x,y)$ ranges over all non-zero integral solutions to $(x,y)\equiv t(a,...
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124 views

Are there any conditions on the moments that make a measure a probability measure?

For a positive Borel measure $\mu$ on the real line interval $[-1, 1]$, let $\displaystyle{m_n = \int_{-\infty}^\infty x^n d\mu(x)}$, i.e. the $n$th moments of the measure. Are there any conditions ...
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27 views

Probability density from standard domain (Typical Box principle and Chinese Remainder Theorem) - III

Pick a random pair $(a,b)\in\mathbb Z_n^2\backslash\{0,0\}$. Denote $N_2(a,b,n)$ to be minimum $\ell_r$ norm of vector $(x,y)$ as $(x,y)$ ranges over all non-zero integral solutions to $(x,y)\equiv t(...
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1answer
70 views

The problems of global asymptotic freeness

Let $X_{N}\in\mathcal{M}_{N}\big(L^{\infty-}(\Omega,\mathbb{P})\big)$ be a $N\times N$ random complex matrix such its entries $(x_{ij}, 1\leq i, j\leq N)$ be $i.i.d.$, centred with variance $1$. $X_{...
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1answer
82 views

What is the pdf of Laplace distribution conditioned on a plane? How can I sample from it?

Our goal is to sample from the Laplace distribution conditioned on a linear subspace. Here are the details of this problem. Let $$p(x) \propto \exp(-\|x\|_1/\sigma)$$ be the pdf of the Laplace ...
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1answer
68 views

Sharp tail bounds for the maximum of an iid sample of a random variable supported on $[0, 1]$

Let $X_1,\ldots,X_n$ be an iid sample from a distribution supported on $[0, 1]$. Question What are some sharp concentration inequalities (i.e tail bounds) empirical statistic defined by $Z_n := \max(...
5
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1answer
60 views

Estimating the size of the remainder in a random partition

Pick a sequence of real numbers $x_i$ as follows. Put $x_0=1$. If $x_i$ is chosen, then pick $x_{i+1}\in[0, x_i]$ according to the uniform distribution. Obviously we have $x_i\rightarrow 0$ with ...
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2answers
53 views

Given two probability density functions find a number that satisfies a given equation

I have a problem for which I either need a proof or a counterexample. We are given two discrete random variables $x_1$ and $x_2$ in $[0, n]$ where $F_1(x)$ is the probability of $x_1\leq x$, and ...
3
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2answers
76 views

What is $\sum_{k=0}^{+\infty}{k⋅p(k;\mu_1,\mu_2)}$, where $p$ is the pmf of Skellam distribution?

The Skellam distribution is the discrete probability distribution of the difference $N_{1}-N_{2}$ of two statistically independent random variables $N_{1}$ and $N_{2}$, each Poisson-distributed with ...
3
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1answer
83 views

What is the expected minimum total matching distance between two partitions of identically and independently distributed points?

Suppose a square $[0,1]\times [0,1]$ in which $N$ vehicles $V_i$ and $N$ riders $R_i$ are distributed identically and independently (say, uniform distribution), a bipartite matching (or a permutation, ...
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0answers
111 views

Calculating the expectation of a sum of dependent random variables

Let $(X_i)_{i=1}^m$ be a sequence of i.i.d. Bernoulli random variables such that $\Pr(X_i=1)=p<0.5$ and $\Pr(X_i=0)=1-p$. Let $(Y_i)_{i=1}^m$ be defined as follows: $Y_1=X_1$, and for $2\leq i\leq ...
3
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2answers
128 views

Expectation of the exitpoint distance for the symmetric random walk

Let $\nu(x)$ be a symmetric probability measure with respect to the origin on $x\in[-1,1]$ such that $\nu(\{0\})\neq 1$. Consider a random walk started at $S_0=0$, denoted $S_n=X_1+\dotsb+X_n$, ...
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0answers
20 views

Non-asymptotic tail-bounds for Hotelling $T^2$ statistic

Let $X_1,\ldots,X_n$ be an i.i.d sample from a distribution on $\mathbb R^p$ with mean $\mu = 0 \in \mathbb R^p$ and $p$-by-$p$ covariance matrix $\Sigma$ of rank $r \le p$. Consider the centered ...
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1answer
67 views

Maximum of sums of iid $X_i$'s where $X_i$ is the difference of two exponential r.v

Given $X_i = A_i - B_i$ where $A_i\sim \text{ Exp}(\alpha)$ and $B_i \sim \text{ Exp}(\lambda)$. Define $S_k = \sum_{i=1}^k X_i$ with $S_0 = 0$, and $$M_n = \max_{1\leq k \leq n} S_k.$$ Is it ...
2
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1answer
202 views

Complicated bound after using Stirling's approximation

I have this inequality $$\frac{1}{a}\exp\bigl\{-\frac{4}{h^2}\bigr\} \geq \frac{1}{f}$$ where $$ a \leq \Bigl(\pi^{d/2}\Gamma(\frac{1}{2}d+1)^{-1} + 1\Bigr) \left(\frac{h^{d+1}}{2} \Gamma \left(\frac{...
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1answer
36 views

Probability of a quantity from Ginibre ensemble

I'm doing a project on random matrices and its applications. I have the joint probability density and want to calculate the probability of $s=\sum_{j=1}^N\lambda_j^2$. So we have $$P(s)=C_{N,K}\int.....
3
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1answer
85 views

Non-asymptotic tail bounds for $D_{\text{Hellinger}}(P\|\hat{P}_N)$

Let P be a distribution on a finite set of size $k$ and let $\hat{P}_N=(N_1/N,\ldots,N_k/N)$ be the empirical distribution (frequencies) from a samples of size $N$. Consider the Hellinger distance ...
1
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1answer
145 views

Deriving condition to get correct asymptotic bound

Suppose that $X\sim \text{Bin}(n,\theta)$. Note that $X$ is the sum of $n$ $iid$ Bernoulli($\theta$) random variables. By the local limit theorem (Theorem 7 here) for the sum of discrete random ...
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1answer
184 views

Mode of a sum of Bernoulli random variables

Let $S_n=\tau_1+\cdots+\tau_n$ be a sum of independent Bernoulli random variables such that $\mathbb{P}(\tau_i=1)=p_i$. Is it true that the mode of $S_n$ is either its mean rounded up or rounded down?
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0answers
45 views

Gaussian as a product of two independent random variables [duplicate]

Ideally what I am looking for two random variables, $X$ and $Y$ (if one is positive then that's even better) such that $Z=X\cdot Y\sim\mathcal{N}(0,1)$ where $X,Y$ are some distributions I can ...
3
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0answers
135 views

Probability distribution from equidistribution - I

Pick a random pair $(a,b)\in\mathbb Z_n^2\backslash\{0,0\}$. Denote $N_r(a,b)$ to be minimum $\ell_r$ norm of vector $(x,y)$ as $(x,y)$ ranges over all non-zero integral solutions to $(x,y)\equiv t(a,...
3
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1answer
114 views

Probability density from standard domain - I

Pick $x+iy$ at random with respect to hyperbolic measure from $\{z:|z|\geq1,|\mathcal R(z)|\leq\frac12\}$. What does the probability distribution function of $\frac1{\sqrt y}$ look like?
3
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2answers
140 views

Is the covariance of squares always bounded from below by two times the covariance?

I came across the following inequality in one of my calculations ($X,Y$ are centered random variables): $$\operatorname{E}(X^2Y^2)-\operatorname{E}(X^2)\operatorname{E}(Y^2) \geq 2 \operatorname{E}(...
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0answers
69 views

Distribution of dot product of two unit complex random vectors [duplicate]

Consider $u,v∈S^{M-1}\subset \mathbb{C}^M$ to be two independent unit norm random vectors on the $M−1$ dimensional complex sphere $S^{M−1}$. In addition, $u$ follows an isotropic distribution, i.e., $...
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2answers
123 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 ...
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0answers
40 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(\...
2
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0answers
106 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 $(X_k)_1^\infty$ ...
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0answers
62 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 ...
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0answers
67 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) ...
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1answer
68 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?
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1answer
43 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, ...
3
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2answers
238 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'}...
2
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1answer
183 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 ...
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0answers
65 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
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1answer
145 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\...
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0answers
54 views

Families of distributions with a certain symmetry property?

Consider the probability distribution $\mathcal{N}_n$ on $\mathbb{R}^n$ whose density is $$(2\pi)^{-n/2}e^{-\frac{1}{2}||\vec{x}||^2} = \prod_{i=1}^n \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x_i^2}$$ ...
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4answers
346 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
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3answers
204 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-...
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2answers
144 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
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1answer
73 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 \...
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1answer
56 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
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1answer
119 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
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1answer
188 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
120 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
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1answer
74 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
315 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 ...
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0answers
51 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
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0answers
132 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
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2answers
97 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_{...