**7**

votes

**7**answers

673 views

### Semicircle law universality elsewhere

Wigner's semicircle distribution is:
$$f(x)=\frac{1}{2 \pi}\sqrt{4-x^2}, \ \ -2\leq x\leq 2.$$
Under reasonable conditions, the rescaled eigenvalue density of random symmetric matrices $M_n$ follows ...

**1**

vote

**0**answers

39 views

### Specifying Skellam parameters by given probabilities

The problem sounds quite easy, and I still think it is. I somehow have the feeling that I just went too far and just miss the easiest solution now. The numerical solution I came up with is just not ...

**2**

votes

**2**answers

137 views

### Are all mixtures of these unimodal functions unimodal?

Let us say that a function $F\colon(0,\infty)\to\mathbb{R}$ is increasing-decreasing if, for some $c\in[0,\infty]$, $F$ is non-decreasing on $(0,c]$ and non-increasing on $[c,\infty)$. Is it true that ...

**0**

votes

**0**answers

85 views

### A question concerning distribution of $\mathbf{Y}/\|\mathbf{Y}\|_2$ where $\mathbf{Y}\sim \mathcal{N}(\boldsymbol{\mu},\mathbf{I})$

I know that when $\mathbf{Y}\sim\mathcal{N}(\mathbf{0},\mathbf{I})$, $\mathbf{Y}/\|\mathbf{Y}\|_2$ is distributed uniformly on the unit sphere. But to my surprise, I failed to find a simple closed ...

**2**

votes

**2**answers

340 views

### Distribution of dot product of two unit random vectors

Consider $\mathbf{u}, \mathbf{v}\in \mathcal{C}^M$ to be two independent unit norm random vectors on the $M-1$ dimensional complex sphere $\mathcal{S}^{M-1}$. In addition, $\mathbf{u}$ follows an ...

**0**

votes

**0**answers

116 views

### Finitely additive measure over integers [duplicate]

We know that, with Axiom of Choice (AC), it can be shown that there exists a finitely additive uniform distribution defined for all subsets of the integers (see, e.g., Hrbacek and Jech 1999, Ch. 11).
...

**1**

vote

**0**answers

49 views

### Lower bound for the probability that a certain component of a Gaussian vector dominates all others

Let $X\sim\cal N(\mu,\Sigma)$ be an $n$-dimensional Gaussian vector. I would like to estimate $$P(X_1>\max_{k=2,\dots,n}X_k).$$
While no closed form solution exists (see e.g. MO question on ...

**0**

votes

**1**answer

83 views

### Finding the distribution of a random variable numerically with sample data? [closed]

Just a thought that I had recently. Suppose given discrete data points for a random variable, could one numerically generate the probability function values at these discrete values? I tried looking ...

**1**

vote

**1**answer

179 views

### Is regularity closed under products?

Let $G \colon [0,1] \to [0,1]$ be a differentiable cumulative distribution function (monotonically non-decreasing function with $G(0) = 0$ and $G(1) = 1$). We say that $G$ is regular if $$ x - ...

**0**

votes

**2**answers

220 views

### How do I Calculate :$\int_{0}^{1}x^{k}\psi(x)dx$ where $k\geq 3$ is an integer?

How do I Calculate, if possible, in terms of well-known constants the integral :
$\int_{0}^{1}x^{k}\psi(x)dx$ , where $k\geq 3$ is an integer ?
note: $\psi(x)$ is digamma function.
Any help would ...

**2**

votes

**2**answers

290 views

### Suggestions for dealing with the “timed” balls-into-bins model

Definitions: Let $T$ (for "time") be a random variable $T \sim \text{Exp}(\lambda)$ and $\Delta t$ is a realization (or called an observed value) of $T$. Let $D$ (for "delay") be a random variable $D ...

**2**

votes

**0**answers

49 views

### Writing a function as a sum of functions of bounded diameter

This problem is distilled from one arising in a study of complex random variables, but I've removed as much baggage as I can without (I hope) making it trivial.
Fix $D>0$. A function $f:\mathbb ...

**1**

vote

**0**answers

44 views

### Is there an equivalent form for Wishart to a power times a normal?

Lin described two equivalent characterizations of the multivariate t-distribution, viz.
As a normal vector divided by an independent chi. That is, $t = Z / \sqrt{\chi^2/v}$, where $Z$ is ...

**0**

votes

**1**answer

60 views

### How can two random variables are continuous infers that their jointly random variable is continuous [closed]

We assume that $\forall a,b$ suchthat $a^2+b^2>0$, $aX+bY$ is continuous random variable.
But we don't assume that $X$ and $Y$ are independent.
My question is the following:
Is it true that the ...

**2**

votes

**1**answer

176 views

### What is the distribution of the maximum nearest-neighbor distance of a point cloud sampled from a solid body like?

Let $\mathcal{B} \subseteq \mathbb{R}^n$ be an $n$-dimensional solid body. Assume that we sample $N$ points, say $S = \{ x_1, ..., x_N \}$, from $\mathcal{B}$ uniformly at random. Consider the ...

**2**

votes

**0**answers

66 views

### expectation involving normal pdf and Rayleigh distribution

I need to calculate following definite integral
\begin{equation*}
\frac{1}{2\pi }\int_0^\infty \frac{x^2 e^{-x^2/\sigma^2 } }{\sigma} \frac{e^{-\frac{\lambda}{{ax^2+b}}}}{\sqrt{ax^2+b}} ~~dx.
...

**2**

votes

**0**answers

57 views

### Compute the smoothing of functions

Given a function $g:R^d\rightarrow R$, which is not necessarily continuous, I want to compute the "smoothing" of $g$, i.e.,
$G(\vec{y})=\int_{R^n} g(\vec{x}) f_{\vec{y}, \sigma}(\vec{x}) d\vec{x} $
...

**4**

votes

**2**answers

348 views

### Expectation of Mahalanobis norm

Let $(g_i)_{i=1,...,d}$ sampled i.i.d. from a standard Gaussian, and $(\lambda_i)_{i=1,...,d}$ non-random s.t. $\max_i(\lambda_i)=1$ and $\lambda_i>0, \forall i$.
I am looking for the expectation ...

**7**

votes

**2**answers

245 views

### A moment problem

Suppose $X, Y$ are two positive random variables such that $\mathbb{E}[X^\alpha] = \mathbb{E}[Y^\alpha]$ for all $\alpha \in (0, 1/2)$.
It is also known that the first moment exists for each of them, ...

**2**

votes

**0**answers

101 views

### Implication of MGF inequality

Let X and Y be two random variables. Denote by $F_X(x)$ and $F_Y(y)$ their CDFs and by $M_X(t)$ and $M_Y(t)$ their MGFs.
It is known that X and Y have the same CDF iff they have the same MGF.
My ...

**5**

votes

**0**answers

131 views

### Elementary function relative to erf

The modified Bessel function of the 1st kind $I_0$ is defined by
$$
I_0(z)=\frac1\pi\int_0^{2\pi}e^{z\cos\theta}\,d\theta
$$
and arises, among other places, in the probability density function of a ...

**2**

votes

**1**answer

106 views

### Variant of Skorokhod's theorem

Consider the following situation:
$S, T$ are standard Borel spaces (say $S = [0,1]^k$, $T = [0,1]$ if it is helpful).
There is a a random variable $\zeta: \Omega \to S$.
$f_n(\zeta) \to^d \eta$, ...

**2**

votes

**0**answers

37 views

### logconcave distribution $f(t)$ leads to concave moments $\mu(x)$. logconvex distribution $f(t)$ leads to convex moments $\nu(x)$?

Let
$$\mu_x=\frac{1}{\Gamma(x+1)}\int_0^{\infty}u^x f(u) du \tag{1}$$
Suppose that $f(u)>0$ when $u>0$ and $f(u)\to 0$ fast enough when $u\to\infty$ so that $\mu_x,-1<x<\infty$ ...

**1**

vote

**1**answer

189 views

### About expectation norms on graphs

Let $S \subseteq V$ of a $d-$regular graph $G$ such that $\mu = \frac{\vert S \vert }{\vert V \vert } $. Let $A$ be the adjacency matrix of the graph. Then define the quantity $\phi(S)= ...

**5**

votes

**2**answers

145 views

### Finding joint probability from double marginals

Consider three probability distributions in the form $p_1(y,z),p_2(x,z),p_3(x,y)$.
When does a global joint probability $p(x,y,z)$ (possibly not unique) exist?
The first compatibility condition to ...

**4**

votes

**2**answers

174 views

### Probable direction of deviations from the expected value in binomial and hypergeometric cases

Suppose I have an urn with N marbles, with frequencies p and q for red and black marbles, and with p > 0,5. I take a sample of r marbles.
It sounds intuitive to say that deviations from the mean ...

**1**

vote

**1**answer

98 views

### A differential inequality and a special value

Let $G \colon [0,1] \to [0,1]$ be a monotonically decreasing function with $G(0) = 1$ and $G(1) = 0$. Suppose that $G$ is differentiable infinitely many times, and that: $$G(x)G''(X) \leq ...

**1**

vote

**2**answers

510 views

### Variance of truncated normal distribution

Let $ X \sim \mathcal{N} ( \mu, \sigma^2 ) $, $ - \infty \leqslant a < b \leqslant +\infty $ ($ a, b \ne \infty $ simultaneously) and $ Y $ has a truncated normal distribution on $ (a, b )$, i.e. ...

**2**

votes

**2**answers

165 views

### Is this a sufficient condition for joint normal distribution?

Suppose I have a random vector $\boldsymbol{Z}$, if I can prove that for $\forall \boldsymbol{\lambda} \neq \boldsymbol{0}$ where $\boldsymbol{\lambda}$ is a fixed vector, not a random vector,
...

**2**

votes

**1**answer

69 views

### Bound on the total variation distance for multiple samples $d_{tv}(P^n,Q^n)$

Given two discrete distributions $P$ and $Q$, with computable total variation distance $d_{TV}(P,Q)=||P - Q||_1$, is there a precise bound for $d_{TV}(P^n,Q^n)=||P^n - Q^n||_1$, as need to estimate ...

**0**

votes

**1**answer

82 views

### Name of distribution [closed]

I am searching for the name of the following distribution on the set of positive integers (including zero).
Let $C\in \mathbb{Z}_+$ and $n\in \mathbb{N}$ are fixed.
Vector $p = (p_1,\ldots,p_n)$ is ...

**5**

votes

**1**answer

1k views

### Square root of normal distribution

Let $X$ and $Y$ be independent random variates with the same probability distribution, $P(x)$. Assuming that the product $Z=XY$ is a random variate with normal distribution, say $$f_Z(x) = ...

**2**

votes

**0**answers

105 views

### Approximate determinantal point process

Consider a random process defined on $2^{\mathcal{X}}$, i.e. all subsets of a set $\mathcal{X}$.
It's well known that this process is determinantal if one can find a positive semidefinite matrix $K$, ...

**2**

votes

**2**answers

182 views

### Does $X_n \xrightarrow{d} N(0,1)$ and $X_n/Y_n \xrightarrow{d} N(0,1)$ imply that $Y_n \xrightarrow{d} 1$?

I'm thinking about the following question:
If $X_n$ and $X_n/Y_n$ both converge in distribution towards a standard Gaussian random variable and $Y_n \geq 0$ for all $n$, does then $Y_n$ necessarily ...

**1**

vote

**0**answers

52 views

### Maximum likelihood estimation with several distributions

My question concerns using Maximum likelihood to estimate unknown parameters used by several (poisson) distributions.
The parameters are the pairs $(a_1,b_1),\dots,(a_N,b_N)$, and for each pair ...

**6**

votes

**2**answers

546 views

### Reference on (discrete) log-concave probability distributions

A discrete distribution $p$ over $\mathbb{N}$ is said to be log-concave if it satisfies the following conditions:
The support of $p$ is a contiguous interval, i.e. $\exists a \leq b$ s.t. $p_i > ...

**3**

votes

**1**answer

212 views

### Lower bound on the tail of the hypergeometric distribution

Suppose there is a bag with $M$ white marbles and $N - M$ black marbles. Let $H(n, N, M)$ be a random variable which is number of white marbles in a draw, without replacement, of $n$ marbles from a ...

**0**

votes

**1**answer

194 views

### Volume of randomly changing sphere follows beta distribution

We are given $X,X_1,\ldots,X_N$ independent and identically distributed $k$-dimensional vectors. For a given query point $X_q\in\mathbb{R}^k$ assume without loss of generality that $X_1,\ldots,X_m$ ...

**4**

votes

**4**answers

347 views

### Central limit theorem with degenerate covariance matrix

Are there known generalisations of the central limit theorem for several random variables when the covariance matrix is degenerate?
The usual proof of CLT based on characteristic functions (see e.g. ...

**1**

vote

**0**answers

50 views

### Shift invariance for the distribution of quadratic polynomials

For a probability distribution $X$, supported on integers, define the shift-invariance of $X$, denoted by $shift(X)$ = total variation distance between the random variable $X$ and $X+1$.
Let ...

**5**

votes

**1**answer

152 views

### Asymptotic behavior of $X_n$ in a Dirichlet vector $(X_1, …, X_n)$

Let $(\alpha_k)$ be a sequence of positive numbers and let $(Y_k)$ be a sequence of independent random variables $Y_k \sim \text{Gamma}(\alpha_k,1)$. Set $X_n=\dfrac{Y_n}{\sum_{i=1}^nY_i}$.
(edit) ...

**2**

votes

**0**answers

224 views

### Inequality with CDF of order statistics

here is a problem I have been struggling with for a while now. This is for a paper I am working on. Any help would be appreciated! Here we go:
Each bidder's valuation $\theta _{i},$ $i=1,...,N$, is ...

**4**

votes

**2**answers

576 views

### Weak convergence of random measures

Let $\mu_n,n\in \mathbb N$ be a random probability measures and let $\mu$ be a deterministic probability measure on $\mathbb R$. That is to say, that the $\mu_n$ are measurable maps from a probability ...

**6**

votes

**0**answers

111 views

### Rate of Convergence of Compound Poisson Laws to Infinitely Divisible Laws

It is known that every infinitely divisible random variable is the limit in law of a sequence of compound Poisson random variables (see for instance Theorem 1.2.18 of Lévy Processes and Stochastic ...

**5**

votes

**1**answer

140 views

### Numerical approximation to the Wasserstein metric?

Are there numerical methods for approximating/calculating the Wasserstein metric in particular cases?
Suppose that $f$ and $g$ are two density functions with the same support. How can I calculate the ...

**3**

votes

**1**answer

80 views

### Error for the convergence by distribution

A sequence of random variables $X_n$ converges in distribution to $X$, if there is pointwise convergence of its characteristic functions, i.e. $\lim_{n\rightarrow\infty}\phi_{X_n}(\lambda) = ...

**0**

votes

**3**answers

279 views

### An inequality based on expectation of continuous random variables

I am trying to prove the following statement:
$$
E[g(X)] E[X^2g(X)]\ge E[Xg(X)] E[Xg(X)]
$$
where $X$ is a random variable, $E[\cdot]$ denotes the expectation operator with respect to ...

**5**

votes

**3**answers

127 views

### Random partitions with prescribed pairwise membership probabilities

Let $(p_{ij}) \in [0,1]^{n \times n}$ be a given symmetric matrix, with $1$ on the diagonal. Suppose $\pi$ is a partition of $[n]=\{1,\dots,n\}$ and let us write $i \stackrel{\pi}{\sim} j$ if $i$ and ...

**1**

vote

**1**answer

282 views

### Convergence in the Wasserstein metric and the square root function

Let $f$ be a smooth probability distribution on the unit square $S$ such that $f(x)>0$ on $S$. Let $\{g_i\}$ be a sequence of smooth probability distributions such that $g_i(x)>0$ on $S$ as ...

**1**

vote

**0**answers

41 views

### A curious example envolving moment's convergence

Let $\{X_n\}$ be a random variable sequence and $X\sim N(0,\sigma)$. In general, the convergence $E(X_n^k) \stackrel{n}{\longrightarrow}E(X^k)$ doesn't implie that $E(X_n^{k+1}) ...