# 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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### 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 : \...

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867 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 $i$.
For ...

**14**

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1k views

### 1-Wasserstein distance between two multivariate normal

The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by
$$d_p(\nu_{1},\nu_{2})=\underset{\pi\in\Gamma(\nu_{1},\nu_{2})}{\inf}\int_{\mathbf{\mathcal{X}}^{2}}d(x,y)^p\pi(dx,dy)$...

**9**

votes

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374 views

### Torus Graph Dynamics

Consider the torus graph, or the toroidal grid, which looks like
(The graph's vertices are the bold dots).
I will discuss only square torus graphs, where there is an equal number of vertices in a "...

**8**

votes

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135 views

### Random arithmetic formulas

An arithmetic formula is a well-formed expression involving only the constant 1, and the binary operations of addition and multiplication, with multiplication by 1 not allowed. For instance, $1 + (1 + ...

**8**

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145 views

### Can one “smooth over” k-wise independence to get actual independence?

I came across the following toy problem and was curious if there was a simple solution or counterexample. Suppose you have a distribution $p$ on $m$ random variables $X_1, \ldots, X_m$, each with ...

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3k 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 $k<d$) ...

**7**

votes

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349 views

### How do I draw samples from this distribution?

Let S be the the standard K-1 simplex. Consider the following probability distribution:
$$\begin{align}
f(p,\alpha,\beta) &= \prod_{k=1}^K p_k^{\alpha_k-1}(1-p_k)^{\beta_k-1}\\
Z(\alpha,\beta) &...

**7**

votes

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558 views

### Product of two random Gaussian matrices - orthant probability

Let $X \in \mathbb{R}^{m \times n}$ and $Y \in \mathbb{R}^{n \times k} $ be two independent Gaussian random matrices, i.e., with entries independently sampled from $\mathcal{N}(0,1)$ (a normal ...

**7**

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133 views

### Joint cumulants of $Z_2^n$ characters

Let $f_{c}:Z_2^n \rightarrow \{-1,1\}$ be the character defined as $f_c(x) = (-1)^{<x,c>}$, where $c,x \in Z_2^n$. It is easy to see that since $f_{c_1}\cdot\ldots\cdot f_{c_k} = f_{c_1 \oplus \...

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256 views

### Calculate the expectation of the maximum of averaged random walks

Let $X_1, X_2, \ldots$ be iid random variables with bounded second moment. The question is to calculate the exact value of $$\mathbb{E} \max_{1 \le j < \infty} \frac{X_1 + \cdots + X_j}{j}.$$
Is ...

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111 views

### Random Balanced Assignment

A balanced assignment from from $N$ objects to $K$ classes is a mapping $\sigma\colon \{ 1, \ldots, N\} \rightarrow \{ 1, \ldots, K\}$ such that
$$
\textrm{Card}( \sigma^{-1} \{j \} ) = \textrm{Card} ...

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147 views

### Distribution of the stopping time of an autoregressive sequence

Consider $e_t$ being i.i.d. uniformly chosen from $\pm 1$. Let $\eta$ be a small positive constant. What is the distribution of $T$ such that $\eta^{0.5} (1+\eta)^T W_T$ first hits $\pm 1$, in which
$$...

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168 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 ...

**6**

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607 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 ...

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**0**answers

389 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 $\frac12$ ...

**6**

votes

**0**answers

172 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 ...

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votes

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1k views

### Hash functions and inner product

As part of a research project on derandomization of linear threshold functions I am working on, I am trying to understand the following problem:
Is there a small (polynomial rather than exponential)...

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168 views

### Total Variation distance of polynomials of Bernoulli R.V.s

Let $X_i, Y_i$ be i.i.d Bernoulli $0/1$ random variables with
$\mathbb{E}[X_i] = p$ and $\mathbb{E}[Y_i] = q$.
Let
\begin{align*}
X &= X_1 X_2 + Χ_2 Χ_3 + \ldots +X_{n-2} X_{n-1}+ X_{n-1} X_n\\...

**5**

votes

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333 views

### How to obtain the probability distribution of a sum of dependent discrete random variables more efficiently

I hope you are well. Here is my problem.
Let $\{s_0,\,s_1,\ldots,\,s_T\}$ be a sequence of discrete random variables and denote $S_t=s_0+s_1+\cdots+s_t$, with $S_0=0$ and $S_T\leq M$, where $M$ and $T$...

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121 views

### anti-concentration of multi-linear polynomials in Gaussian variables

A Gaussian variable $X_i\sim {\cal N}(0,1)$ is anti-concentrated in the following sense: for any $\epsilon>0$ we have:
$$
\mathbf{P}( |X_i| \leq \epsilon ) = O(\epsilon).
$$
Hence if we consider a ...

**5**

votes

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354 views

### Closedness of a set of measures, where conditional marginals are in closed $\varepsilon$-ball w.r.t. Wasserstein distance

Let $(E,d)$ be a bounded polish space (separable, complete metric space satisfying $\sup_{x,y\in E} d(x,y) < \infty$). By $\mathcal{P}(E)$ we denote the space of Borel probability measures on $E$ ...

**5**

votes

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294 views

### integrating with respect to parameters in beta function

I would like to evaluate an integral:
$$\int_t^1\frac{1}{B(1+s\phi,1+\phi(1-s))}p^{s\phi}(1-p)^{\phi(1-s)}ds,$$
where $B(a,b)$ is a beta function and $p\in(0,1)$ and $\phi>0$ are some parameters. ...

**5**

votes

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51 views

### Why is it easy to compute the first and fourth moments for random chord length in a convex solid?

Recently I was led to some considerations in geometric probability, a field pretty far from any specialization of mine. (Context: I was working with a collaborator on a question about mean escape ...

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votes

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115 views

### Distribution of Random Knots from Braids

Let $R_{2n,l}$ be a random braid word of length $l$, where each letter is chosen uniformly from the braid generators of $B_{2n}$, $\{\sigma_1,\ldots,\sigma_{2n-1},\sigma_1^{-1},\ldots,\sigma_{2n-1}^{-...

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231 views

### universality for large deviations?

This is a question about universality in probability theory, with combinatorics in mind.
Consider a sequence of polynomials $P_n$ in one variable, with positive coefficients. Combinatorics is a large ...

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votes

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156 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 ...

**5**

votes

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216 views

### A note on Doob's theorem

I have faced the following problem, regarding to the Martingale Theory. Because this area far from my area I don't know whether this problem is in literature or this can be simple question for ...

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votes

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500 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 ...

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266 views

### References for this game

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 probability ...

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votes

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857 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. ...

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votes

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110 views

### Under what conditions is conditional expectation a bijective operator of distributions

Suppose that we have two independent random variables $V$ and $W$ over $\mathbb{R}$. Suppose that $W$ has a probability density with respect to the Lebesgue measure.
My question: Can we find ...

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votes

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78 views

### Asymptotics of the joint pdf of two sums of powers of independent $\mathcal U(0,1)$ random variables

As a warm-up in words: The sum of twelve uniform random variables is a classic approximation to a normal distribution. What is the joint pdf for the sum of their cubes and the sum of their fourth ...

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votes

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207 views

### Minimization over a convex function of equal vs unequal success probabilities of Bernoulli random variables

Let $U_1,U_2,\ldots,U_n$ be $n\geq 2$ mutually independent Bernoulli random variables. There are two cases of interest:
$1.$ The random variables $U_1,U_2,\ldots,U_n$ are identically distributed;
$...

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votes

**0**answers

73 views

### What is the entropy of binomial decay?

Let's play a game. I start with $N$ indistinguishable tokens, and I wait $T$ turns. Every turn, each token has probability $p$ of disappearing. I want an analytic formula for the entropy of this ...

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votes

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314 views

### Sum of Binomial random variable CDF

Suppose there are two independent Binomial random variables
$$
X\sim Binomial(n,p)\\
Y\sim Binomial(n,p+\delta)
$$
where $\delta$ is considered to be fixed, and $p$ can vary in $(0,1-\delta)$.
Now ...

**4**

votes

**0**answers

77 views

### Does a non-exchangeable empirical reverse-martingale exist?

Consider a possible finite sequence $\xi_1,\xi_2,\dots$ of random variables and consider the measure-valued empirical process
$$\eta_n=\frac{\sum_{i=1}^n\delta_{\xi_i}}{n},\:\:\: n=1,2,\dots$$
Assume $...

**4**

votes

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86 views

### Concentration Inequality for Score Functions of Exponential Familty

Let $p$ be the density of a continuous one-parameter exponential family distribution on $\mathbb{R}$. We assume that
$$p(x) = c(x)\cdot \exp\bigl [ x \cdot \theta - b(\theta ) \bigr ], $$
where $\...

**4**

votes

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87 views

### Spectrum of sum of fixed matrices with random signs

Let $A_1,\ldots,A_k$ be a given sequence of $N$-by-$N$ Hermitian matrices. Assume all have spectrum contained in $[-1,-\delta] \cup [+\delta,+1]$ for some $\delta>0$. Let $$A=\frac{1}{\sqrt{k}} \...

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129 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 $i$.
...

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226 views

### Under what conditions do time averages of ergodic transformations satisfy a central limit theorem?

Let $(X, \mu)$ be a probability space and $T:X\rightarrow X $ an ergodic transformation, i.e. $T$ is measure preserving and the only $T$ invariant subspaces have either measure $0$ or measure $1$ (...

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votes

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690 views

### Total variation and Hellinger distance inequality between truncated Gaussians

We know that the total variation distance, $d_{TV}(P,Q) = \frac{1}{2}\left|\left|P-Q\right|\right|_1$, between any two distributions $P$ and $Q$ is lower bounded by their squared Hellinger distance, $...

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174 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 $i,j$, i....

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votes

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72 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) = \...

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votes

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207 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 integer-...

**4**

votes

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302 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, ...

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782 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 ...

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votes

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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 ...

**3**

votes

**0**answers

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,...

**3**

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83 views

### Probability distributions with all positive cumulants

Is there a term for a distribution with all cumulants positive (or nonnegative)?