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.
1,922
questions
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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&...
6
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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 ...
6
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1
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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.
...
6
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1
answer
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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 ...
6
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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 ...
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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 ...
6
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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 \...
6
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1
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232
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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 ...
6
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1
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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 ...
6
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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$...
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answer
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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$ ...
6
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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}^\...
6
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1
answer
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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 ...
6
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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 ...
6
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0
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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 $S=X_1+\cdots+X_n$....
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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 ...
6
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131
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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
$$
\textrm{Card}( \sigma^{-1} \{j \} ) = \textrm{Card} ...
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Closedness of a set of measures, where conditional marginals are in closed $\varepsilon$-ball w.r.t. Wasserstein distance
Let $(E,d)$ be a bounded polish space (separable, complete metric space satisfying $\sup_{x,y\in E} d(x,y) < \infty$). By $\mathcal{P}(E)$ we denote the space of Borel probability measures on $E$ ...
6
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Distribution of the stopping time of an autoregressive sequence
Consider $e_t$ being i.i.d. uniformly chosen from $\pm 1$. Let $\eta$ be a small positive constant. What is the distribution of $T$ such that $\eta^{0.5} (1+\eta)^T W_T$ first hits $\pm 1$, in which
$$...
6
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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 $f$. ...
6
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0
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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 ...
6
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0
answers
199
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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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0
answers
466
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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
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0
answers
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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 ...
6
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0
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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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2
answers
544
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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 ...
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2
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880
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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 ...
5
votes
1
answer
886
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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 ...
5
votes
3
answers
594
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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 ...
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2
answers
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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 ...
5
votes
2
answers
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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 ...
5
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3
answers
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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 ...
5
votes
3
answers
632
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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 ...
5
votes
2
answers
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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 ...
5
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2
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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. ...
5
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2
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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 ...
5
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2
answers
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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 ...
5
votes
2
answers
273
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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 ...
5
votes
1
answer
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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 ...
5
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2
answers
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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 $\...
5
votes
1
answer
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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?
...
5
votes
2
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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 $f\geq 0, \int_{0}^{1} f(x) dx = 1$...
5
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2
answers
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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
$$\pi\delta(0)=...
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2
answers
138
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Number of resampling until obtaining a uniform list
Let $A_0$ be a list of $ n$ distinct elements. By sampling with replacement the elements of $A_0$, we obtain a new list $A_1$ of $n$ elements that are not necessarily distinct. Repeat the same process ...
5
votes
2
answers
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Limit of the extremal process of i.i.d. Gaussians see from the tip
I'm trying to calculate the weak limit of $\mathcal{E}_N(x)=\sum_{k=1}^{2^N}\delta_{x-Z_k}$ , with $Z_k=X_k-\max_{k\leq 2^N}X_k$, $\{X_k\}$ being $2^N$ copies of i.i.d. Gaussians with mean zero and ...
5
votes
1
answer
999
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Quantization of normal distribution
For $n\in\mathbb{N}$, denote by $\mathcal{Q}_n$ the set of all probability measures on $\mathbb{R}$ that are supported on at most $n$ points.
Question: Is it known which element in $\mathcal{Q}_n$ is ...
5
votes
4
answers
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The distance distribution of graphs
The degree distribution of a graph is of main importance, especially for large graphs, and namely random graphs. Its expected value and its higher moments tell a lot about a graph – but of course not ...
5
votes
1
answer
837
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Moments of maximum of independent Gaussian random variables
Let $X = (X_1, \ldots, X_d) \in \mathbb{R}^d$ be a mean-zero Gaussian random vector with identity covariance matrix. Are there upper bounds for
$$E \left(\|X\|_{\infty}^k \right)$$ for $k=1, \ldots, ...
5
votes
2
answers
695
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Negative probabilities - what are two ordinary pgfs that correspond to the gf of a half-coin?
In Half of a Coin: Negative Probabilities, author considers pgf of a fair coin represented by random variable, $X = 1_H$:
$$G_X(z) = E[z^X] = \sum_{x=0,1} z^xP(X=x) = (z^0)(1/2) + (z^1)(1/2) = \frac{...
5
votes
1
answer
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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 $...