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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Ordering preference for two zero mean Gaussian outcomes

Let $X\sim \mathcal{N}(0,1)$ be a standard Gaussian random variable. If we let $f_a(x)\triangleq\mathbb{E}[\max\{aX,x\}]$ for $a,x >0$, how to prove that $$f_a(f_b(1))<f_b(f_a(1))~~\text{for }0&...
Pierre's user avatar
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Probabilistic Proofs of Key Number-Theoretic Results

Given a positive integer $n$, let $p$ be the largest prime less than or equal to $n$. Let $N(n)=2^{C_2}\cdots p^{C_p}$ be uniformly distributed from $1$ to $n$, and $M(n)=2^{Z_2}\cdots p^{Z_p}$ where ...
The Substitute's user avatar
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1 answer
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Convergence of Radon Nikodym derivatives

I apologise in advance if my question is too basic. Some notation: $(X,\cal{X})$ denotes a measurable metric space where $X$ is a metric space and $\cal{X}$ is the associated Borel sigma algebra. ...
Eduardo's user avatar
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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 ...
Mehmet Ozan Kabak's user avatar
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Probability distributions: The maximum of a pair of iid draws, where the minimum is an order statistic of other minimums?

General question: What is the distribution for the maximum of 2 independent draws from cdf F(x), when we know that the minimum of those same two draws is the kth order statistic of the minimum of n ...
OctaviaQ's user avatar
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2 answers
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Marginal distribution of the diagonal of an inverse Wishart distributed matrix

This is a cross-posting of a question I asked at CrossValidated. It hasn't generated much activity so I'm trying here: Suppose $X\sim \operatorname{InvWishart}(\nu, \Sigma_0)$. I'm interested in the ...
JMS's user avatar
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3 answers
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Curvature function as a random variable with uniform distribution

Let $(S,g)$ be a Riemannian surface. Then the curvature function $\kappa: S\to \mathbb{R}$ can be counted as a random variable. So it produces a probability density function $f_g:\mathbb{R}\to \...
Ali Taghavi's user avatar
6 votes
1 answer
232 views

Which orthant probabilities are the largest? (For a multivariate normal distribution)

I have a $k$-dimensional multivariate normal distribution $X∼N(0,\Sigma)$ with covariance matrix $\Sigma$. $\Sigma$ has two distinct eigenvalues, say $\lambda_1 > \lambda_2$, with orthogonal ...
Matthew Harrison-Trainor's user avatar
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1 answer
377 views

Functional limit theorem under random change of time

FINAL EDIT: There is one main question left: According to the answer, we have choosen $\theta=1$ , where we could choose $0<\theta<\infty$ as we like. His this sufficient, if we regarde the ...
GuildY123's user avatar
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1 answer
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Properties of a finite random walk

Consider the simplest random walk - $X_0 = 0$ and from there on (i.i.d), $X_i=X_{i-1}+1$ with probability $p$ or $X_{i-1}-1$ otherwise. Let $Y_N$ be the highest point $X$ have reached on the first $N$...
R B's user avatar
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Stein's Lemma for Discrete Distribution

Stein's Lemma in its standard form states that $X \sim N(0,1) \Leftrightarrow E[f'(X) - X f(X)] =0 $ for all bounded one-time differentiable functions $f$ (I am ignoring the exact conditions on $f$ ...
Kcafe's user avatar
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1 answer
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Summability of ratios of moments a weight

Recently, I encounter the following problem: Let $w$ be a probability density on $[0,1]$. Let mk be the $k$-th moment, i.e., $$m_k=\int_0^1t^kw(t)dt.$$ Under what condition can we have $$\sum_{k=0}^\...
Yanqi QIU's user avatar
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How to check if a symmetric random variables is the difference of two iid symmetric random variables

I have the continuous symmetric random variable $X$ in $\mathbb{R}$. If I know its distribution function $F(x)$ what are the conditions on $F(x)$ so that $X=Y_1 - Y_2$ where $Y_i$ are also iid ...
Jose M. Del Castillo's user avatar
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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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Distribution of iid hypergeometric random variables conditioned on the sum

Let $X_1,X_2,\ldots,X_n$ be iid random variables with hypergeometric distribution. To be specific, $$ \mathrm{Prob}(X_1=i) = \frac{\binom{N}{i}\binom{M-N}{m-i}}{\binom{M}{m}}.$$ Let $S=X_1+\cdots+X_n$....
Brendan McKay's user avatar
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Delayed Pólya's urn process

The standard Pólya's urn process can be stated as follows: You have an urn with red and green balls. At any time unit you choose one ball at random, note the colour, and give the ball back. At the ...
Matjaž Krnc's user avatar
6 votes
0 answers
131 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} ...
Steve's user avatar
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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 $E$ ...
Steve's user avatar
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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 $$...
Minkov's user avatar
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6 votes
1 answer
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Choosing a sample based on where the density function is highest

Is there a name for the following process? Say I have an absolutely continuous probability density function $f$ with compact support, and I take $k$ independent samples $x_1,\dots,x_k$ from $f$. ...
Tom Solberg's user avatar
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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 ...
F. C.'s user avatar
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6 votes
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199 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 ...
Bjørn Kjos-Hanssen's user avatar
6 votes
0 answers
466 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 $\frac12$ ...
Brendan McKay's user avatar
6 votes
0 answers
185 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 ...
user22624's user avatar
6 votes
0 answers
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)...
Guy Adini's user avatar
5 votes
2 answers
544 views

A measure of how "spread out" a probability measure is

Consider a random variable $X$ whose variance is large. As a contrast to Markov's or Chebyshev's inequality, both of which measure the concentration of a probability distribution, is there a measure ...
Snoop Catt's user avatar
5 votes
2 answers
880 views

Subgaussian norm of a symmetric $\{-1,0,1\}$ random variable

Let $p\in [0,1/2]$, and define $\xi$ as the symmetric random variable such that $$ \xi = \begin{cases} 1 & \text{ w.p. } p\\ 0 & \text{ w.p. } 1-2p\\ -1 & \text{ w.p. } p \end{cases} $$ so ...
Clement C.'s user avatar
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5 votes
1 answer
886 views

How can I sample uniformly from a citrus surface?

I want to sample from a Lemon surface uniformly. The equation of this surface is $$16(x^2+z^2)+(y-2)^3 y^3=0.$$ I have read the paper Stratified Sampling of 2-Manifolds . The method described in this ...
haotian yi's user avatar
5 votes
3 answers
594 views

Convergence speed of a random dyadic rational generator

We are given a multiset $M$ of real numbers which initially is equal to $\{0,1\}$. In a sequential fashion, at each round $r\in\mathbb{N}$ two distinct instances $x_r$ and $y_r$ of $M$'s numbers are ...
Penelope Benenati's user avatar
5 votes
2 answers
421 views

An interesting calculation of derivative

I was trying to get the probability distribution $p(n)$ from a generating function $G(s)$ like this: $G(s) = e^{a(s-1)^2}=\sum s^np(n)$ I need first to do Maclaurin expansion of the exponential and ...
doubllle's user avatar
  • 153
5 votes
2 answers
7k views

Upper bound total variation by Wasserstein distance for continuous distance

I am reading the survey of the relationships between metrics of distributions (see https://arxiv.org/pdf/math/0209021.pdf for the paper). The general results show that for general distributions, we ...
Felix Y.'s user avatar
  • 123
5 votes
3 answers
4k views

Distribution of the individual coordinates of a uniform random vector on a high-dimensional sphere

Let $X=(X_1,\ldots,X_n)$ be a random vector uniformly distributed on the $n$-dimensional sphere of radius $R > 0$. Intuitively, i think that for large $p$ every coordinate $X_i$ is normally ...
dohmatob's user avatar
  • 6,716
5 votes
3 answers
632 views

The relative error of approximating a binomial

Are there any good approximations for a binomial CDF that work well in terms of the relative error, as opposed to absolute? For the usual normal approximation, the absolute error is very well-studied ...
Tom Solberg's user avatar
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5 votes
2 answers
176 views

Density near at $0$ for the integral of the positive part of the Brownian motion

This question was asked recently on MO and then deleted by the owner, user Aalon. I think the question deserves to be answered, which is what I will try to do here. Aalon was reading this paper, where ...
Iosif Pinelis's user avatar
5 votes
2 answers
2k views

low-discrepancy sequences for sampling of distributions

Hi everybody, I am writing of physics simulation that traces charged particles. I need to set up these particles in phase space ( 3 space dimension + 3 momentum dimensions) following distributions. ...
Radfahrer's user avatar
5 votes
2 answers
255 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 ...
geodude's user avatar
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5 votes
2 answers
1k views

The distribution of balls in a Bean Machine that omits all the "prime pegs"?

The Bean Machine, also known as a quincunx or the Galton box, is a well known triangular board that contains several rows ($n$) of staggered, but equally spaced pegs. Balls are dropped from the top ...
Agno's user avatar
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5 votes
2 answers
273 views

A comparison of diffusions

Consider two diffusions given by $$X_j(t)=\int_0^t a_j(s,X_j(s))\,dW_s$$ for $j=1,2$ and $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and the $a_j$'s are smooth enough ...
Iosif Pinelis's user avatar
5 votes
1 answer
261 views

Random domino tilings: Is this distribution well-defined, and how can it be sampled from?

I'd like to ask questions about a "random domino tiling of the plane". However, it's not quite obvious how to go about precisely specifying what this means. My first instinct was to do ...
RavenclawPrefect's user avatar
5 votes
2 answers
2k views

Tight sequence of measures

This is probably a very easy question for experts in probability or measure theory. I have a sequence of finite measures $\mu_{n}$ on a non-compact metric space $X$ such that $\mu_{n}$ converges to $\...
AMath91's user avatar
  • 38
5 votes
1 answer
3k views

Eigenvalues and eigenvectors of Gaussian random matrices

Let us assume we have a square matrix $A$ whose entries are sampled from a standard Gaussian distribution of mean $0$. Do we have any information about the distribution of its eigenvalues? ...
Alfred's user avatar
  • 879
5 votes
2 answers
792 views

Can the differential entropy of a continuous distribution with lebesgue integrable density be negative infinity?

Define the (differential) entropy for a density $f$ as $$ H(f) :=-\int_{0}^{1} f(x) \log_{2}(f(x)) dx \, .$$ I am trying to find a $f \in L_{1}([0,1])$ such that $f\geq 0, \int_{0}^{1} f(x) dx = 1$...
Tejas Bhojraj's user avatar
5 votes
2 answers
772 views

Was there ever proposed a theory where the value of Dirac Delta at zero had meaning on itself?

Was there ever proposed a theory where $\delta(0)$ has a meaningful value or used in a formal way outside integrals? Particularly, following Fourier transforms, we can formally obtain $$\pi\delta(0)=...
Anixx's user avatar
  • 9,306
5 votes
2 answers
138 views

Number of resampling until obtaining a uniform list

Let $A_0$ be a list of $ n$ distinct elements. By sampling with replacement the elements of $A_0$, we obtain a new list $A_1$ of $n$ elements that are not necessarily distinct. Repeat the same process ...
minhtoan's user avatar
  • 1,404
5 votes
2 answers
179 views

Limit of the extremal process of i.i.d. Gaussians see from the tip

I'm trying to calculate the weak limit of $\mathcal{E}_N(x)=\sum_{k=1}^{2^N}\delta_{x-Z_k}$ , with $Z_k=X_k-\max_{k\leq 2^N}X_k$, $\{X_k\}$ being $2^N$ copies of i.i.d. Gaussians with mean zero and ...
MikeG's user avatar
  • 665
5 votes
1 answer
999 views

Quantization of normal distribution

For $n\in\mathbb{N}$, denote by $\mathcal{Q}_n$ the set of all probability measures on $\mathbb{R}$ that are supported on at most $n$ points. Question: Is it known which element in $\mathcal{Q}_n$ is ...
Steve's user avatar
  • 1,085
5 votes
4 answers
2k views

The distance distribution of graphs

The degree distribution of a graph is of main importance, especially for large graphs, and namely random graphs. Its expected value and its higher moments tell a lot about a graph – but of course not ...
Hans-Peter Stricker's user avatar
5 votes
1 answer
837 views

Moments of maximum of independent Gaussian random variables

Let $X = (X_1, \ldots, X_d) \in \mathbb{R}^d$ be a mean-zero Gaussian random vector with identity covariance matrix. Are there upper bounds for $$E \left(\|X\|_{\infty}^k \right)$$ for $k=1, \ldots, ...
Kcafe's user avatar
  • 509
5 votes
2 answers
695 views

Negative probabilities - what are two ordinary pgfs that correspond to the gf of a half-coin?

In Half of a Coin: Negative Probabilities, author considers pgf of a fair coin represented by random variable, $X = 1_H$: $$G_X(z) = E[z^X] = \sum_{x=0,1} z^xP(X=x) = (z^0)(1/2) + (z^1)(1/2) = \frac{...
BCLC's user avatar
  • 237
5 votes
1 answer
2k views

expected value of multiplication of matrices

I start with background and then ask my question, background is a brief description of wishart distribution. Background The Wishart distribution with $\nu$ degrees of freedom and positive definite $...
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