The probability-distributions tag has no usage guidance.

**2**

votes

**2**answers

83 views

### splitting exponential random variable into independent components

$X$ follows Exponential $(\lambda)$. Can we split $X$ into two independent r.v.'s, i.e.,
do there exist functions $g$ and $h$ such that $g(X)$ and $h(X)$ are independent for any fixed $\lambda$? ...

**5**

votes

**1**answer

122 views

### Estimating entropy conditional to an event

Take for example the measure $\mu(n)=n^2$ on $\{1, \ldots, N\}$ and a random variable $X$ distributed according to the probability obtained by normalizing $\mu$.
Does there exists a constant ...

**3**

votes

**1**answer

48 views

### Reference request for a result regarding density of induced probability measure under a submersion

Let $\pi: M \to N$ be a smooth submersion from a bounded open subset of $\mathbb{R}^m$ onto $ N \subset \mathbb{R}^n$, $m \geq n$. Further, let $M$ be given a probability measure $\mu$. Then the map ...

**4**

votes

**1**answer

99 views

### Zeta zeros standard normal distribution about $\vartheta (\gamma_n)$

Asked at MSE here without response.
I realise that this resembles Odlyzko's famous nearest neighbours plot, and was wondering whether this is simply a manifestation of the same phenomenon.
That ...

**0**

votes

**0**answers

28 views

### How to determine if random variables are distributed according to a multivariate normal distribution? [migrated]

Suppose $(x_1, x_2, x_3)\sim N(\mu, \Sigma)$ where $\mu\in\mathbb{R}^3$ and $\Sigma$ is a $3\times 3$ covariance matrix, are the variables $A = x_1 + x_2$ and $B = x_2 + x_3$ necessarily distributed ...

**0**

votes

**0**answers

58 views

### Is this probability distribution studied in literature?

Let $\theta_1,\theta_2,\theta_3$ be 3 non-negative random variables such that $\theta_1+\theta_2+\theta_3=1$ with the joint probability distribution
\begin{align}
...

**1**

vote

**0**answers

18 views

### What is the Blumenthal-Getoor index of Student's distributions?

For infinitely divisible random variables, Blumenthal and Getoor introduced in [1] an index that allow to study for instance the local Hölder regularity of Lévy processes. For an infinitely divisible ...

**1**

vote

**2**answers

72 views

### The Levy measure of the compound Poisson distribution

The compound Poisson distribution is defined as(see Levy processes and infinitely divisible distributions page: 18):
Let $c>0$ and $\sigma$ be a measure on $\mathbb{R}$ with $\sigma(\{0\})=0$, a ...

**0**

votes

**0**answers

51 views

### Conditions for Mellin inversion

Under which conditions is the function
$$
g(s)=a^{c(s-1)}\Gamma(s),\qquad a>0,c\in \mathbb{R}
$$
the Mellin transform of a probability density function $f$? If $c=-1$, then $f$ is the exponential ...

**0**

votes

**0**answers

30 views

### Log-concave distributions: Weighted sum of pdfs

Assuming $f_n(\cdot)$ is a log concave function (e.g., pdf of Gaussian distribution) and $0\le q_n\le 1$ for all $n\le N$, I am trying to find conditions under which the following holds
...

**0**

votes

**0**answers

28 views

### Interpolating a polynomial when we permute part of $y_i$'s

Let $\vec{x}=[x_1,...,x_n]$ be elements of field $\mathbb{Z}_p$, where $p$ is a large prime. $x_i \neq x_j$, $x_i \in \mathbb{Z}_p$. Note $x_i$ values are NOT picked uniformly random and they are ...

**2**

votes

**1**answer

82 views

### Orthogonal polynomials with respect to the lognormal distribution

I am currently doing some inspection on the orthogonal polynomials with respect to the lognormal distribution. Does anyone already work on that or know some cool references?
All the best,
Pierre-O.

**0**

votes

**0**answers

37 views

### Quotient of cumulative binomial distribution functions

Given to integers $n < m \in \mathbb{N}_0$ and a probability $p$, I'm struggling to calculate (or at least get an upper bound for) the quotient
$$Q = \frac{F(n+1;m,p)}{F(n;m,p)}$$
where $F$ denotes ...

**5**

votes

**1**answer

65 views

### when does elementwise-log preserve positive-semidefiniteness?

Let $Z$ be a positive semidefinite matrix with nonnegative entries, and define $X=\log(1+Z)$, where the $\log$ is taken entrywise, i.e., $X_{ij}=\log(1+Z_{ij})$. Are there some simple sufficient ...

**0**

votes

**0**answers

35 views

### Validating a probability density distribution forecast model for a Markov process

Let's say we have a Markov process $X_t$, and we come up with a forecast model that takes some information from outside world and says: "value $X_{t+1}$ has probability density distribution $P_t(x)$". ...

**0**

votes

**0**answers

40 views

### $\exists \mathcal{A},\mathcal{B}:X\sim \mathcal{A}\Rightarrow \frac{p}{\sqrt{q+rX}}\sim \mathcal{B}$?

Does there exist a parametric distribution $\mathcal{A}$, such that:
$X\sim \mathcal{A}\Rightarrow\frac{p}{\sqrt{q+rX}}\sim \mathcal{B}$ for some parametric distribution $\mathcal{B}$
Where ...

**1**

vote

**0**answers

52 views

### What is a two-sided geometric distribution?

I found in some articles (such as this) references to two-sided geometric distribution. But I went through texts of probability and did not find anything called "two-sided geometric distribution". ...

**2**

votes

**0**answers

40 views

### A canonical example of the non-existence of predictive probability distribution

Section 3 of Fortini et al. (2000) states that
Given $(X^\infty, \mathcal X^\infty,P)$, a predictive probability distribution of $x_n$ given $(x_1, \dots, x_{n-1})$ with respect to $P$ need not ...

**0**

votes

**0**answers

85 views

### Probability of substring given string production probabilities

I originally posted this question on the Math StackExchange, but have not received answers there and thought it might be more appropriate to post it here.
Let $\Sigma$ be an alphabet and let $y = x_1 ...

**4**

votes

**0**answers

93 views

### Dimension reduction for low-order moments of Rademacher-weighted sums of vectors

Let $x_1,\dots,x_n$ be vectors in a Euclidean space $H$. Let $\varepsilon_1,\dots,\varepsilon_n$ be independent Rademacher random variables (r.v.'s), so that $P(\varepsilon_i=\pm1)=1/2$ for all $i$.
...

**-2**

votes

**1**answer

52 views

### expected value of cosine wirh Gaussian phase

Is there a solution to the expected value/variance for a Gaussian with random phase:
$$\cos(\omega_0 t + \phi), \qquad \phi \sim \cal{N}(0,\sigma^2) $$
?
For $t=0$, the solution is for example ...

**1**

vote

**0**answers

23 views

### PDF of points at the intersection of a sphere and hyperboloid in n dimensions

I'm studying a statistical mechanics problem and I have two conserved quantities:
$$ E = \sum_{k=0}^{M} \left[ a_1^2(k) + a_2^2(k) + b_1^2(k) + b_2^2(k)\right] $$
$$ H = \sum_{k=0}^{M} 2 k \left[ ...

**6**

votes

**1**answer

124 views

### Is there an $\infty$ version of the Wasserstein distance between two distributions?

If I have two probability distributions $\mu$ and $\nu$ defined on $X$ and $Y$ respectively, then the $p$-th Wasserstein distance between the two of them is defined as $$W_p(\mu,\nu) = ...

**3**

votes

**1**answer

104 views

### Maximal entropy distribution with given conditionals

It is well known that of all the joint distributions $p(x,y)$ with fixed marginals $p(x),p(y)$, the one with the highest entropy is:
$$
p(x,y)=p(x)p(y).
$$
Suppose instead that we have conditionals. ...

**7**

votes

**1**answer

154 views

### Distribution of infinity-norm over the unit sphere

I need to compute probabilities of the form
$P( \Vert X \Vert_\infty < r ),$
where $X$ is a random variable of dimension $n$, drawn with a uniform distribution on the unit sphere ...

**0**

votes

**0**answers

82 views

### Conditional probability of dependent random variables

Let $ X \sim f_X(x), Y \sim f_Y(y) $ are two dependent random variables and their corresponding PDFs. I want to find a probability $$ P(Y\ge 0 | X+Y\ge 0) .$$ If these variables were independent I'd ...

**5**

votes

**0**answers

136 views

### Extrapolation between longest increasing and longest alternating subsequences

The question
When should we expect Tracy-Widom?
motivated me to post the following question, in which I have been
interested for a while. Let $f(n)$ be a function from the positive
integers to ...

**13**

votes

**2**answers

404 views

### A probability distribution in n dimensional space which its projection on any line is a uniform distribution?

Does there exist, for any natural $n$, a probability distribution in $\mathbb{R}^n$ whose projection on any line is a uniform distribution?

**0**

votes

**0**answers

76 views

### Why is this distribution exponential?

Take the interval $[0, 1]$.
Now sample 10000 points in this interval randomly according to the uniform distribution.
The fact is that the distribution of the distances between adjacent points on ...

**2**

votes

**0**answers

27 views

### Terminology for research on distributions of inner products

Consider a set of vectors $M$ from an inner product space $V$. The ordered set of inner products of all pairs of elements in $M$ uniquely characterizes $M$ up to isomorphism.
Suppose now that $V$ is ...

**7**

votes

**7**answers

593 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

24 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

132 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

76 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 ...

**0**

votes

**0**answers

69 views

### Probabilities in a directed graph

Given a directed graph of "n" vertices, having on average "m" out-edges each, what is the probability that an arbitrarily chosen vertex will belong to a unique circuit?
Also, how does that ...

**0**

votes

**0**answers

84 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

114 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

40 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

62 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

171 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

189 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

280 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

41 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

30 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

53 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

148 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

47 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

53 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

310 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

230 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, ...