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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1answer
23 views

Maximum of bounded expectations at a certain Borel set?

Assume ${\bf x} \in \mathbb{R}^n$ denotes a real-valued bounded random variable with a distribution measured on the Borel space $(\mathbb{R}^n,\mathcal{B}^n)$. Let $f:\mathbb{R}^n\to\mathbb{R}$ denote ...
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1answer
30 views

Ensemble averaging in a random graph (or network) in the large $N$ limit

I have a random graph/network described by the adjacency matrix $(a_{ij})_{N\times N}$ where $a_{ij}=1$ with probability $p$. Each node in the graph is associated with a continuous quantity $\eta_i=\...
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1answer
60 views

Explicit constant for Carbery–Wright inequality

The Carbery-Wright is a seminal result about the anti-concentration of polynomials of Gaussian random variables. See e.g. https://arxiv.org/pdf/1507.00829.pdf, Theorem 1.4, for the precise statement. ...
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How to calculate Chi-Square density value only known P-value? [migrated]

Everywhere online there is how to calculate the Chi-Square density value given a confidence level: $\alpha$/p value; but I can not find how one calculates the inverse? How to calculate the $\alpha$/p-...
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Probability that a Voronoi cell contains exactly k random points

Gilbert's argument [1962] for a given point to be contained in a Voronoi cell of area $s$ is that, known the p.d.f. of cell areas -- be it $f(s)$ --, then the probability of $f(s|X=1)=sE[s]^{-1}f(s)$ ...
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1answer
26 views

Are p-boxes for discrete sample spaces meaningful?

The Wikipedia article about p-boxes only talks about cumulative probability density functions, which are meaningful for continuous sample spaces. https://en.wikipedia.org/wiki/Probability_box Just ...
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27 views

Average number of edges of an induced graph by using the edge-based node selection technique on a graph with arbitrary degree distribution

Let $G(U,V,E)$ be a simple, undirected, bipartite graph and $U=\{u_1,u_2,{\cdots},u_n\}$ and $V=\{ v_{1},v_{2},\cdots,v_{n}\}$. Let $d_k^l$ be the number of vertex with degree $k$ in $l$, where $l \in ...
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1answer
89 views

Conditional expectation values defined by expectation values

I asked this question a couple of days ago on Math.SE but without any echo (no upvotes, although I offered a bounty). But because I did it for oversight from a reputable/professional source I now ...
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0answers
37 views

Harmonic mean or generating function, pick one?

Is there an obvious or a profound reason for the dearth of statistical distributions for which analytical harmonic mean and moment generating functions co-exist? Are there good examples (with finite ...
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1answer
77 views

Finding a connection between two types of convergence

Please, help me find connections between two types of convergence: Let $\{X_n\}_{n\ge1}: (\Omega,F,P) \rightarrow (\mathbb{R},Bor)$ be a sequence of r.v., there are two convergences: 1) $X_n \...
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1answer
114 views

Is this probability inequality true?

This question may be simple, though I'm not managing to find an answer. Let $X$ and $Y$ be two dependent random vectors in in $\mathbb{R}^d$, with joint probability density $\mu(x,y)$ (with respect to ...
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3answers
77 views

Reshaping a Gamma random variable?

Suppose that $X \sim \Gamma(\alpha_1,1)$, a random variable with gamma distribution with shape $\alpha$ and unit rate/scale. Q: Can we found a reshaping function $f_{\alpha_1\rightarrow\alpha_2}$ ...
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A medicine heals 60% of the patients [closed]

A medicine heals 60% of the patients. 10 sick people get this medicine. a) What is the probability that all 10 people are cured? b) What is the probability that at least 1 of the people is cured? ...
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How to avoid using a probability distribution that doesn't exist?

I have this problem, of which I know the solution, but I'm looking for the mathematically proper way of writing it. Say I have a (infinite) population of people, where each individual is labeled by ...
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0answers
70 views

Weak convergence and Lipschitz function

I want to construct such r.v. ${ξ_n}$, $n≥1$$: (Ω,F,P)→(R^1,Bor)$, ${ξ_n}$ weakly converges to ${ξ}$ and such $f$ - Lipschitz function , so that $E(|f(ξ_n)−f(ξ)|) \not\to0$. I tried to apply ...
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2answers
100 views

Expected value of a truncated binomial

Let $X\sim B(n,p)$ be a binomial random variable and fix $0<k<n$. Are there any well-known bounds for $\mathbb{E} (X-k)^+$, where $(X-k)^+ =\max\{0,X-k\}$? I am particularly interested in ...
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1answer
44 views

Stein's lemma for Gaussian variables proof

I am reading a paper (https://arxiv.org/abs/1001.3448) and they mentioned Stein's lemma (below) as a useful fact without proof, I also read the reference in the paper but I got nothing. Please help me ...
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1answer
50 views

A uniform mixture of order statistics

Let $0<k<n$ be integers, and let $X$ be a random variable obtained as follows: sample $n$ points independently and uniformly at random in the unit interval, and select (uniformly) one of the $k$...
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2answers
102 views

Disintegration, conditional probabilities, and conditional expectation

On the Wikipedia page there is a note that conditional probability measures can be described by disintegration. However, I can seem to find a clear exposée of how this construction is related to ...
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1answer
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Can the joint law $P \circ (X,Y)^{-1}$ of two random variables $X$ and $Y$ be written as $P \circ (X,\phi(X,U))^{-1}$ for $U$ uniform in $[0,1]$?

I want to know whether there is some general assumpitons we can make on two measurable spaces $E$ and $F$ (e.g. polish, complete, separable,...) such that we can ensure that the following "Theorem" ...
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42 views

Concentration (or two sided tail bounds around expectations) of maximum and minimum of $n$ iid, subgaussian random variables

I asked this on MSE, but got no answer, hence asking here now. Help appreciated! My question is motivated by this question and this question, where the first was aimed for giving a one sided tail ...
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1answer
62 views

The weak convergence of finite dimensional distribution of Gaussian process does not imply the weak convergence in $C[0,1]$

In the study of weak convergence in $C[0,1]$, a common example is always being considered: $$X_{n}(t)=nt1_{[0,1/n]}(t)+(2-nt)1_{(1/n,2/n]}(t).$$ This example serves a counter-example to show that the ...
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1answer
58 views

Find a conditional expectation of a difference of two independent Poisson process

Consider two independent Poisson processes $N,M$ with rate $\lambda$, and define $$X(t):=x+\dfrac{1}{\sqrt{n}}[N(t)-M(t)].$$ From this formula we know that $X(0)=x$. Now, I want to compute the ...
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1answer
121 views

Average number of elements of a subset S of a matrix A after inducing the rows and columns of m randomly selected elements from subset S

Let $A_{N{\times}N}$ be an $N{\times}N$ matrix and $\mathcal{S_{k}}$ be a subset of elements in $A$ such that exactly $k$ elements from every row and column in $A$ are in $\mathcal{S_{k}}$. Thus, $\...
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1answer
26 views

Lyapunov condition for CLT for asymptotically independent sequence

Suppose I have some triangular array $\{X_{n,j}\}$ of random variables, which need not be independent or identically distributed. Suppose I further know that $$Var\left(\sum_{j=1}^n X_{n,j}\right)\to \...
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0answers
43 views

Berry-Esseen unit ball

How can I show that sampling a random vector from a uniform distribution over a $d$-dimensional unit ball is similar to sampling a random vector from a uniform distribution over $d$-dimensional ...
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258 views

Analyzing my definition of Average which uses a variation of the Lebesgue Integral and Measure [closed]

Consider $f:A\to[a,b]$ where $A\subseteq[a,b]$ and $S\subseteq A$. As noted in previous questions, I want to define an average using a new measure and integral since I found certain aspects of the ...
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55 views

Manifold structure of Gaussian mixtures

Fix $l$ a positive integer. Let $\mathcal{M}$ denote the set of Gaussian mixtures of the form $$ \sum_{i=1}^l k_i \mu_i, $$ where $\mu_i $ is a non-degenerate Gaussian measure on $\mathbb{R}^k$ and ...
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0answers
32 views

Metrics on the space of distributions in terms of p.d.fs

If two probability distributions (on the same measure space) are s.t they have p.d.fs and the $L^1$ distance between the p.d.f.s is large, then is there a choice of a ``nice" metric $d_{\rm smooth}$ ...
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36 views

Can we write down the density of this distribution?

Simple version: I am looking for the density of the random vector $(X+Z,Y+Z)$, where $X,Y,Z$ are independent gamma random variables (with non-restricted parameters). Next step: Actualy, i'm looking ...
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1answer
49 views

Concentration of norm of linearly transformed normal random vector as dimension go to infinity

Earlier asked on MSE, but didn't get an answer, so posting here: Let $X=(X_1 \dots X_n) \in \mathbb{R}^n, X_i\sim N(0,1), iid.$ Let $B: \mathbb{R}^n \to \mathbb{R}^n $ be the diagonal linear map: $...
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1answer
105 views

Comparing $X+Y$ and $X-Y$ for independent random variables with values in an abelian locally compact group

Let $G$ be an abelian locally (separable?) compact group with Haar measure $\mu$. Inspired by the interesting proof of A sum of two binomial random variables : Let $X$ and $Y$ be $G$-valued ...
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0answers
43 views

What is the distribution of the norm of the multivariate $X \sim \mathcal{N}(\mu, \Sigma) \in \mathbb{R}^d?$

Let $X \sim \mathcal{N}(\mu, \Sigma) \in \mathbb{R}^d$ follow a multivariate normal distribution. Then what's the distribution (PDF, CDF etc.) of $X?$ When $\mu = 0, \Sigma = I_d,$ we know that $||X||...
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0answers
61 views

Dependence rank: what is the size of the largest subcollection of random variables which is statistically independent?

Let $X_1,\ldots,X_p$ be random variables on the same space. Define their dependence rank, denoted $rank(X_1,\ldots,X_p)$ as the largest nonnegative integer $k$ such that there is a subcollection of $k$...
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0answers
39 views

Comparing Euclidean norm of two normal vectors

Let $X_i$ ($i = 1,2$) be two random vectors in $\mathbb R^n$, with normal distribution with scalar covariance matrix $\sigma_i^2$ and center $\mu_i$ (in my case, $n = 2$). Is there a way to estimate ...
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1answer
136 views

Inequality on the Hellinger distance between Poisson and mixture of Poisson

Let $H$ denote the Hellinger distance; i.e., for two discrete distributions $p,q$ (identified with their pmf) over $\mathbb{N}$, $$ H(p,q)^2 = \frac{1}{2}\sum_{n=0}^\infty \left(\sqrt{p(n)}-\sqrt{q(n)}...
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1answer
75 views

Integral rising from difference of chi-squared random variables

Let $X,Y$ be independent random variables such that $X\sim\chi_{n-1}^{2}, Y\sim\chi_{1}^{2}$ are chi-squared distributed (where $n\geq2$ is a natural number). I am trying to evaluate $\mathbb{P}[X\leq ...
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1answer
50 views

fractional moments of multivariate normal distributions

is there an analytic formula for fractional moments of multivariate normal distribution? $E(\prod_{i=1}^k x_i^{\nu_i})=?$ where $X=(x_1,\ldots,x_k) \sim N_k(\mu, \Sigma)$, $\nu_i\in \mathcal{R}$ and $\...
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1answer
92 views

Asymptotically tight concentration of norms of subgaussian random vectors with independent coordinates, as the dimension $n \to \infty?$

Let $X=(X_1 \dots X_n)\in \mathbb{R}^n,$ be a subgaussian random vector so that $X_i$'s are independent, $\mathbb{E}X_i = 0, \mathbb{E}X_i^2=1.$ Before we pose our question, let's state the following: ...
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0answers
25 views

What is the conjugate prior of Multivariate Log Normal distribution?

For the univariate log-normal distribution, when mean is known, the conjugate prior is gamma distribution. But how about the multivariate log-normal distribution ? What is the conjugate prior ? ...
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0answers
37 views

Concentration of $2$-norms of random variables whose co-ordinates are not independent?

Let us consider the random vector $X=[X_1 \dots X_d] \in \mathbb{R}^d, E[X]= 0, cov[X]= \Sigma.$ Then the random vector $Z:= \Sigma^{-1/2} X=[Z_1 \dots Z_d]$ has $E[Z]=0, cov[Z]=I_d.$ I'm looking for ...
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0answers
55 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 ...
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0answers
36 views

How to take into account the properties of $M(n)$ numbers and improve the variance of normal distribution?

It is known the normal approximation to $inv(\pi)$: $$ \left| P\left( \frac{\mathrm{inv}(\pi)-\frac 12{n\choose 2}}{\sqrt{n(n-1)(2n+5)/72}}\leq x\right)-\Phi(x)\right| \leq \frac{C}{\sqrt{n}}, $$ ...
3
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1answer
131 views

Prove or disprove the linearity of expectiles

For expectation (mean), there are many useful properties such as Linearity of Expectation: $\mathbb{E}[X+Y]=\mathbb{E}[X]+\mathbb{E}[Y]$ $\mathbb{E}[\alpha X]=\alpha\mathbb{E}[X]$ (The 2 equations ...
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1answer
45 views

Variational problem: how to minimise the second moment?

This is a neater version of a question I posted here, on which I'm also stuck. The problem: Say I have a probability density function $f(x)$, defined for positive $x$, and let's note its $n$th non-...
2
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1answer
86 views

An elementary question on probability distributions

I have encounter the following problem, but after trying a little I did not arrive to a good conclusion. Suppose that $X$ is a positive random variable for which we only know that $E[X] = 2$ and $E[1/...
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0answers
44 views

Upper bound on the condition number of the product of a random sparse matrix and a semi-orthogonal matrix

Let $G \in \mathbb{R}^{n \times m}$ (m > n, m = O(n)) whose all entries are i.i.d. distributed as $\mathcal{N}(0, 1) * \text{Ber}(p)$. Let $V \in \mathbb{R}^{m \times n}$ be a fixed semi-orthogonal ...
4
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3answers
97 views

Find distribution that minimises a function of its moments

Imagine a probability density function $f(x)$, defined for positive $x$, and let's note its $n$th non-centred moment $x_{n}$. The mean $x_{1}$ is fixed (and positive). How can I find $f(x)$ that ...
2
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1answer
111 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? ...
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2answers
56 views

Random variable corresponding to sum of density functions [closed]

The distribution of functions of random variables is well-studied for various different and general cases, but I didn't find much result for the reverse. Suppose that $X_1, X_2$ are (probably ...

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