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14
votes
0answers
362 views

Erdos-Kac for squarefree numbers

In its usual form, the Erdos-Kac Theorem states that if $f(n) : \mathbb{N} \rightarrow \mathbb{R}$ is a strongly additive function with $|f(p)| \le 1$ for all primes $p$, then $$\frac{\\#\{n \le x : ...
12
votes
0answers
531 views

On random Dirichlet distributions

Fix a dimension $d\ge2$. Let $Q_d$ denote the positive quadrant of $\mathbb{R}^d$, that is, $Q_d$ is the set of points $\mathbf{x}=(x_i)_i$ in $\mathbb{R}^d$ such that $x_i>0$ for every $i$. ...
7
votes
0answers
227 views

Inequality between incomplete beta and gamma functions; or when is binomial distribution function above/below its limiting Poisson

Please note: this question was posted first (September 4) in math.stackeschange.com and then (September 16) in stats.stackeschange.com. It got no answers in neither of those sites. Let the ...
5
votes
0answers
205 views

1-Wasserstein distance between two multivariate normal

The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by ...
5
votes
0answers
103 views

Two sets of independent Bernoulli random variables

There are two sets of random variables $X_1,\ldots,X_n$ and $Y_1,\ldots,Y_n$ satisfying: Each $X_i$ and each $Y_j$ has a symmetric Bernoulli distribution ($-1$ and $+1$ with probability $\frac12$ ...
5
votes
0answers
140 views

what books to read to quickly understand adiabatic approximation

Hi group, I'm a theoretical ecologist with fairly adequate training in applied math (ODE, linear algebra, applied probability, some PDEs). In my current work, I've encountered the use of adiabatic ...
5
votes
0answers
361 views

Compute the expected value of the next step of a sorted random walk

Here's what I'm thinking about. If you have a random walk (move +1 or -1 at each step) of some fixed length, then if you're at the maximum of the walk, the next step you take is -1 with probability 1. ...
4
votes
0answers
53 views

Cramér-Wold theorem with independence assumption

Let $X = (X_1, X_2)$ be a random vector with joint probability density $p$. The celebrated Cramér-Wold theorem says that we can reconstruct $p$ from knowing the push-forward densities of $X$ under all ...
4
votes
0answers
45 views

Is there a name for the set of distributions whose probability generating functions are Mobius transformations?

Consider a discrete random variable $N\in\mathbb N$ with $\mathbb P(N=0) = p$, $\mathbb P(N=n) = (1-p)(1-q)q^n$ for $n\neq 0$. Then the probability generating function of $N$ $$\mathbb E(z^N) = ...
4
votes
0answers
169 views

probabilistic terminology for polynomials with positive coefficients

Given a polynomial $P(x) = p_0 + p_1 x + p_2 x^2 + ... + p_n x^n$ with non-negative coefficients, is there a standard name for (the function of $p_1,...,p_n$ equal to) the variance of an ...
4
votes
0answers
266 views

When is taking an average (mean) an algebraic operation in the sense of monads?

Taking the average of a sequence of numbers is not an "algebraic" operation, in the following sense. Given sequences $X_1,X_2,\ldots,X_n$ of numbers, one could either take the average of each one, ...
3
votes
0answers
108 views

Concentration inequality for function of independent Bernoulli r.v.'s (related to random graph)

Consider a random undirected graph on a set of $n$ nodes, say $\{1,2,\ldots,n\}$, such that the probability of edge between nodes $i$ and $j$ is $p_{ij}$ (we may assume $p_{ij}=o(1)$ for all $i,j$, ...
3
votes
0answers
651 views

E[ | X - Y | ] where X and Y are independent Poisson random variable

What is the expected value of the absolute difference of two independent Poisson variables? E[ |X - Y| ] Seems like an easy question but I haven't found an easy solution. I've split the double sum ...
3
votes
0answers
180 views

References for this game

Hello everybody, I would like to know how the following game is known in the literature and, possibly, to have references for related papers. Description of the game: Fix a space $X$ and two Borel ...
3
votes
0answers
275 views

Is this probability distribution known in the literature?

In some work I was doing I derived a probability distribution that I do not recognize. Is it a known distribution? $\Pr(X\le ...
3
votes
0answers
143 views

Iterated Kumaraswamy distributions

The Kumaraswamy distribution has cdf $F(x;a,b) = 1-(1-x^a)^b$. Does anyone know any formulas or properties relating to iterations of this on itself, meaning $$ F_i(x;a,b) = 1-(1-F_{i-1}^a)^b$$ If ...
2
votes
0answers
52 views

Can truncated/non-smooth distributions be used as priors/posteriors in Variational Bayesian methods?

Variational Bayesian methods can sometimes be a good alternative to Markov Chain Monte Carlo numerical evaluation of probability distributions. They do this, as I understand it, by approximating the ...
2
votes
0answers
46 views

Is this exponential family Gaussian?

Let $m$ be a positive measure on the real line. Assume that $\exp k(t)=\int\exp(xt)m(dx)<\infty$ for $a<t<b$ and that $\exp k(t)=\infty$ if $t\notin[a,b],$, with $-\infty\leq a<b\leq ...
2
votes
0answers
35 views

existence of Markov operators not generated by transition probability function

Transition probability functions can always be used to generate Markov operators, correct? So is it correct to say that a Markov process is a collection of Markov operators? On the other hand, are ...
2
votes
0answers
82 views

Quantifying the “flatness” of functions which are the Fourier transforms of positive functions

Short version of question: I'm trying to understand the extent to which a function is prevented from being "flat" as a result of being the Fourier transform of a positive function. That is, the extent ...
2
votes
0answers
93 views

Marginalizing multivariate normal over defined interval

Hello everyone, I am trying to obtain an analytic expression for the following Gaussian integral $$\frac{1}{\sqrt{(2 \pi)^n |\Sigma|}} \int \kern-0.2em \cdots \kern-0.2em \int d\mathbf{x}_{\sim i} ...
2
votes
0answers
43 views

Minimizing/Maximizing the tail of the convex combinations of Chi Squared i.i.d random variables

Consider $N$ i.i.d random variables, $X_{1}, X_{2}, \ldots, X_{N}$ , that are chi-squared of degree $K \geq 2$. Also consider the following 3 vectors: \begin{eqnarray*} \bar{a} &=& ...
2
votes
0answers
149 views

Have you seen this one parameter family of distributions before?

This is a one parameter family of distributions. Choose some parameter $\lambda > 0$ and define the measure $\nu_\lambda$ which is absolutly continuous with respect to the Lebsegue measure with the ...
2
votes
0answers
104 views

question about circular law

Hi, I have a question about the circular law. Consider $A_n=[x_{ij}]$ a sequence of random matrices where $x_{ij}$ are iid with mean $0$ and variance $1$. Consider $\lambda_{n,1},\dots,\lambda_{n,n}$ ...
2
votes
0answers
184 views

Proving that an increasing iterative sequence increases at a decreasing rate

In this question Proving a sequence of integrals increases (iterated minimax distributions) Pietro Majer proved that $$\int_0^1F_n(x) dx \leq \int_0^1F_{n+1}(x) dx$$ when $$F_n(x) = ...
2
votes
0answers
140 views

Secretary problem extensions

In the Classical Secretary problem (also known as Marriage, Sultan's Dowry, Gogol problems), 1) There are n candidates ordered from the best to the worst (no ties). We know n. 2) The candidates ...
2
votes
0answers
258 views

Seeking the normalizing constant (or any references) for a distribution over a subset of positive definite martrices

I'm interested in a probability distribution over the set of positive definite matrices with unit diagonal elements. That is, and $X$ such that: $X \in S^{n+}, \forall_{i}X_{ii} = 1$ where $S^{n+}$ ...
2
votes
0answers
273 views

Computable distribution on [0,1] with C-infinity distribution function

Does anyone know of an easily-describable distribution on $[0,1]$ with a density $p$ (with respect to Lebesgue measure) that satisfies the following properties: $p$ is $C^\infty$ $p(0) = a$, $p(1) = ...
2
votes
0answers
520 views

For what sub-$\sigma$-algebra are these two measures equivalent?

In two statistics papers (linked inline below) I have come across two definitions of certain probability measures. I conjecture that for particular choices of the construction that they are ...
1
vote
0answers
33 views

Angular distribution for Gaussian vector with non-zero mean

The angular central Gaussian distribution (ACG) is the distribution of $\frac{\mathbf{x}}{\|\mathbf{x}\|}$, when $\mathbf{x}\sim\mathcal{N}\left(\boldsymbol{0},\mathbf{A}\right)$, where $\mathbf{x}$ ...
1
vote
0answers
24 views

Conditional Distribution of Inverse Wishart

Suppose $\begin{bmatrix} K_{11} K_{12}\\K_{12}^T K_{22} \end{bmatrix}\sim\mathcal{IW}\left(\eta,\begin{bmatrix} \Sigma_{11} \Sigma_{12}\\\Sigma_{12}^T \Sigma_{22} \end{bmatrix}\right)$. What is the ...
1
vote
0answers
36 views

Does this kind of integral equations have unique solution?

Suppose $f_1$ and $f_2$ are two probability density functions on support $[0,1]$ (i.e. $f_1(x)=f_2(x)=0$ for any $x\not\in[0,1]$). Let $\varphi(x)$ denote a known probability density function on ...
1
vote
0answers
49 views

Distribution of the local time for reflected Brownian motion on the quadrant

If $W$ and $B$ are two independent Brownian motions, $X_t=W_t-\sup_{u\le t} B_u + 1$. I want to find the distribution of $S$ the first hitting time of $0$ by $X$. If someone could give me a direction ...
1
vote
0answers
101 views

approximation of probability distribution

I have a question: Let $\mu$ be a probability distribution defined on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ satisfying $$\int_{\mathbb{R}}|x|d\mu<+\infty$$ Set $$A_n=\Big\{\frac{i}{n}:~ ...
1
vote
0answers
57 views

Cramér-Wold like theorem for independent random variables

Let $X$ be a random vector in $\mathbb R^n$ with probability distribution $\mathbb P_X$. Now when given only the family of distributions \begin{align*} \left\{ \mathbb P_{v_1 X_1 + \dots + v_n ...
1
vote
0answers
46 views

Characteristic function known on subsets

Let $X$ be a random variable in $\mathbb R^n$ with distribution $\mathbb P_X$. Given a (infinite) family of matrices $W_t \in \mathbb R^{n \times m}$ parameterized by $t \in \mathbb R$, suppose we ...
1
vote
0answers
124 views

Sampling a two-dimensional Gaussian distribution at points along an integer lattice

Please consider a two-dimensional Gaussian of the general form: $A*e^{-(\frac{(x-x_0)}{2\sigma_x^2}+\frac{(y-y_0)}{2\sigma_y^2})}$, where $C$ is the peak of the Gaussian, i.e. the point at which the ...
1
vote
0answers
59 views

Multiple independent random number streams

This question is somehow related to this one. Having multiple streams of pseudo-random numbers known to be independent and with a uniform distribution (U1, U2,...,Un) I want to do Monte Carlo ...
1
vote
0answers
50 views

Moments of Matrix Gamma distribution

Matrix gamma distribution (defined for example in http://en.wikipedia.org/wiki/Matrix_gamma_distribution) is one way to generalize Wishart distribution. In our course work that distribution was used ...
1
vote
0answers
71 views

Sum of non-identical categorical random variables

Is there a named distribution for the sum of non-identical categorical random variables? When the categorical variables are i.i.d., the sum is a multinomial distribution. When the categorical ...
1
vote
0answers
115 views

Nontransitive dice: the least number of faces?

Here is an introduction to nontransitive dice. The question is: given $n$-player with a $m$-sided dice each one, the what is the minimum of $m$ for a fixed $n$ to produce nontransitivity? Here is ...
1
vote
0answers
162 views

Joint distribution from multiple marginals

Consider an experiment consisting of a repeated trial with two random Bernoulli (=binary) variables, A and B. Each trial consists of multiple outcomes for both A and B. Each trial has the same number ...
1
vote
0answers
323 views

Prove that the sum of a certain infinite series is 1

Prove the (numerically-evident) proposition that \begin{equation} \Sigma_{i=0}^\infty f(i) = 1, \end{equation} where \begin{equation} f(i)= 2^{-4 i-6} q(i) \frac{\Gamma(3 i+\frac{5}{2}) \Gamma(5 ...
1
vote
0answers
92 views

Inequality for square of the subgaussian distributions

Hi all, For my research I am trying to bound some exponential moments of subgaussian r.v.'s. And I am stuck with proving one of such inequalities. More specifically: Let $a$ be unit vector in ...
1
vote
0answers
520 views

Distribution of Inverse of a Random Matrix

Recently i got stuck into a problem and couldn't find its satisfactory answer anywhere. My question is simple. Suppose i have a fat random matrix (i,e $R$ has dimensions $k\times d$ where $k<d$) ...
1
vote
0answers
234 views

Distribution of random vectors

Two positive numbers $\alpha$ and $\beta$ are given. We are going to describe a process of choosing a random vector on the unit sphere $S$ in $\mathbb R^3$ (given by $x^2+y^2+z^2=1$). A vector $u\in ...
1
vote
0answers
170 views

Why this two model have same probability distribution?

(1) Consider the following method of generating a random tree with $n$ nodes. First expand the root node into two branches. Then expand one of the two terminal nodes at random. At time $k$, ...
1
vote
0answers
385 views

Can you prove the monotonicity of the function (or find a counter example)?

Let $X$ be a non-negative random variable that is drawn from a cumulative distribution function $F(\cdot)$, pdf $f(\cdot)$ and mean $E[x]$. $k$, $c$, $v_l$ and $v_h$ $(v_h>v_l)$ are non-negative ...
1
vote
0answers
155 views

Non-decreasing probability of detecting the most-likely event from a multinomial distribution

I examine a multinomial distribution with parameters $\\vec{p} = [p_1, p_2, \\ldots, p_s]$. I denote the cells by $C = \{ c_1, c_2, \\ldots, c_s\}$. In this setting $p_{[i]}$ is the $i^{\\text{th}}$ ...
1
vote
0answers
107 views

What is the maximum a Pearson's chi square statistic can get?

I actually know that the answer is $N(k-1)$ (where $k$ is the minimum between number of rows and number of columns). However, I can not seem to find a simple proof for why the statistic is bounded ...