# Tagged Questions

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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### 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 ...
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### 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 \begin{...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 $P_t(x)$". ...
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### 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 $i$. ...
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### 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 ...
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### 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. ...
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### 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 $\mathcal{S}_{n-1}$...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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). ...
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### 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 ...
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### 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 ...
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