# 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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### 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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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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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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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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**1**answer

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(...

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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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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 ...

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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 ...

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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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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 ...

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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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### 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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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 ...

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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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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.....

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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 ...

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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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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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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 ...

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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,...

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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?

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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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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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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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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(\...

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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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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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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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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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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, ...

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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'}...

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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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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 ...

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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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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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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 ...

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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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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 ...

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**1**answer

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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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:
$\...

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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 ...

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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 ...

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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 $l$-...

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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 ...

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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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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 ...

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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_{...

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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_{...