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.
1,264
questions
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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 ...
-2
votes
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=\...
1
vote
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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0answers
9 views
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-...
3
votes
0answers
26 views
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 $f(s|X=1)=sE[s]^{-1}f(s)$ ...
0
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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 ...
0
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0answers
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 ...
1
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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 ...
0
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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 \...
0
votes
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 ...
1
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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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0answers
27 views
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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0answers
81 views
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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votes
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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
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$...
1
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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 ...
0
votes
1answer
24 views
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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0answers
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 ...
1
vote
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).$$ This example serves a counter-example to show that the ...
0
votes
1answer
58 views
Find a conditional expectation of a difference of two independent Poisson process
Consider two independent Poisson processes $N,M$ with rate $\lambda$, and define $$X(t):=x+\dfrac{1}{\sqrt{n}}[N(t)-M(t)].$$ From this formula we know that $X(0)=x$. Now, I want to compute the ...
2
votes
1answer
121 views
Average number of elements of a subset S of a matrix A after inducing the rows and columns of m randomly selected elements from subset S
Let $A_{N{\times}N}$ be an $N{\times}N$ matrix and $\mathcal{S_{k}}$ be a subset of elements in $A$ such that exactly $k$ elements from every row and column in $A$ are in $\mathcal{S_{k}}$. Thus, $\...
0
votes
1answer
26 views
Lyapunov condition for CLT for asymptotically independent sequence
Suppose I have some triangular array $\{X_{n,j}\}$ of random variables, which need not be independent or identically distributed. Suppose I further know that
$$Var\left(\sum_{j=1}^n X_{n,j}\right)\to \...
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votes
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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votes
0answers
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 ...
0
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0answers
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 ...
0
votes
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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0answers
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 ...
0
votes
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: $...
5
votes
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 ...
0
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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 $k$...
0
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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 ...
3
votes
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)}...
0
votes
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 ...
2
votes
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 $\...
1
vote
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 ? ...
0
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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 ...
4
votes
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 ...
0
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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
votes
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 ...
1
vote
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
votes
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/...
0
votes
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
votes
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
votes
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?
...
0
votes
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 ...
