The probability-distributions tag has no wiki summary.

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### Expectation with respect to empirical distribution [on hold]

Let $(\Omega,\mathcal{A})$ be a measure space and $X$ a random variable with distribution $P$. The expectation of some measurable function $g$ with respect to $P$ is
$$
\mathbb{E}_P[g(X)] = ...

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38 views

### Compute the smoothing of functions

Given a function $g:R^d\rightarrow R$, which is not necessarily continuous, I want to compute the "smoothing" of $g$, i.e.,
$G(\vec{y})=\int_{R^n} g(\vec{x}) f_{\vec{y}, \sigma}(\vec{x}) d\vec{x} $
...

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257 views

### Expectation of Mahalanobis norm

Let $(g_i)_{i=1,...,d}$ sampled i.i.d. from a standard Gaussian, and $(\lambda_i)_{i=1,...,d}$ non-random s.t. $\max_i(\lambda_i)=1$ and $\lambda_i>0, \forall i$.
I am looking for the expectation ...

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214 views

### A moment problem

Suppose $X, Y$ are two positive random variables such that $\mathbb{E}[X^\alpha] = \mathbb{E}[Y^\alpha]$ for all $\alpha \in (0, 1/2)$.
It is also known that the first moment exists for each of them, ...

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71 views

### Implication of MGF inequality

Let X and Y be two random variables. Denote by $F_X(x)$ and $F_Y(y)$ their CDFs and by $M_X(t)$ and $M_Y(t)$ their MGFs.
It is known that X and Y have the same CDF iff they have the same MGF.
My ...

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11 views

### Variance of the maximum likelihood estimator of Rayleigh Distribution [migrated]

I want to calculate the variance of the maximum likelihood estimator of a Raileigh distribution using $N$ observations.
The density probability function of this distribution is :
$$
f(\sigma,y_i) = ...

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113 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 ...

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97 views

### Variant of Skorokhod's theorem

Consider the following situation:
$S, T$ are standard Borel spaces (say $S = [0,1]^k$, $T = [0,1]$ if it is helpful).
There is a a random variable $\zeta: \Omega \to S$.
$f_n(\zeta) \to^d \eta$, ...

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34 views

### logconcave distribution $f(t)$ leads to concave moments $\mu(x)$. logconvex distribution $f(t)$ leads to convex moments $\nu(x)$?

Let
$$\mu_x=\frac{1}{\Gamma(x+1)}\int_0^{\infty}u^x f(u) du \tag{1}$$
Suppose that $f(u)>0$ when $u>0$ and $f(u)\to 0$ fast enough when $u\to\infty$ so that $\mu_x,-1<x<\infty$ ...

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86 views

### Upper bound involving random orthogonal projection

Let $R$ be an $n\times N$ random matrix with i.i.d. standard Gaussian entries, $n<N$, and let $M:=(RR^T)^{-1/2}R$. Let $u,v\in \mathbb{R}^N$ non-random and s.t. $u^Tv=0$ and $\|u\|>\|v\|$.
I ...

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166 views

### About expectation norms on graphs

Let $S \subseteq V$ of a $d-$regular graph $G$ such that $\mu = \frac{\vert S \vert }{\vert V \vert } $. Let $A$ be the adjacency matrix of the graph. Then define the quantity $\phi(S)= ...

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92 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 ...

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161 views

### Probable direction of deviations from the expected value in binomial and hypergeometric cases

Suppose I have an urn with N marbles, with frequencies p and q for red and black marbles, and with p > 0,5. I take a sample of r marbles.
It sounds intuitive to say that deviations from the mean ...

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83 views

### A differential inequality and a special value

Let $G \colon [0,1] \to [0,1]$ be a monotonically decreasing function with $G(0) = 1$ and $G(1) = 0$. Suppose that $G$ is differentiable infinitely many times, and that: $$G(x)G''(X) \leq ...

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170 views

### Variance of truncated normal distribution

Let $ X \sim \mathcal{N} ( \mu, \sigma^2 ) $, $ - \infty \leqslant a < b \leqslant +\infty $ ($ a, b \ne \infty $ simultaneously) and $ Y $ has a truncated normal distribution on $ (a, b )$, i.e. ...

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138 views

### Is this a sufficient condition for joint normal distribution?

Suppose I have a random vector $\boldsymbol{Z}$, if I can prove that for $\forall \boldsymbol{\lambda} \neq \boldsymbol{0}$ where $\boldsymbol{\lambda}$ is a fixed vector, not a random vector,
...

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27 views

### Bound on the total variation distance for multiple samples $d_{tv}(P^n,Q^n)$

Given two discrete distributions $P$ and $Q$, with computable total variation distance $d_{TV}(P,Q)=||P - Q||_1$, is there a precise bound for $d_{TV}(P^n,Q^n)=||P^n - Q^n||_1$, as need to estimate ...

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58 views

### Name of distribution [closed]

I am searching for the name of the following distribution on the set of positive integers (including zero).
Let $C\in \mathbb{Z}_+$ and $n\in \mathbb{N}$ are fixed.
Vector $p = (p_1,\ldots,p_n)$ is ...

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100 views

### Log-concavity of convolution log-concave and not log concave density functions

Let $Z = X +Y $ be a random variable (r.v.) where $X$ and $Y$ are independent r.vs. If the density function of $X$ and $Y$, $f(x)$ and $g(x)$ are log-concave in the support of $X$ and $Y$, ...

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229 views

### Square root of normal distribution

Let $X$ and $Y$ be independent random variates with the same probability distribution, $P(x)$. Assuming that the product $Z=XY$ is a random variate with normal distribution, say $$f_Z(x) = ...

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54 views

### Approximate determinantal point process

Consider a random process defined on $2^{\mathcal{X}}$, i.e. all subsets of a set $\mathcal{X}$.
It's well known that this process is determinantal if one can find a positive semidefinite matrix K, ...

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169 views

### Does $X_n \xrightarrow{d} N(0,1)$ and $X_n/Y_n \xrightarrow{d} N(0,1)$ imply that $Y_n \xrightarrow{d} 1$?

I'm thinking about the following question:
If $X_n$ and $X_n/Y_n$ both converge in distribution towards a standard Gaussian random variable and $Y_n \geq 0$ for all $n$, does then $Y_n$ necessarily ...

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39 views

### Maximum likelihood estimation with several distributions

My question concerns using Maximum likelihood to estimate unknown parameters used by several (poisson) distributions.
The parameters are the pairs $(a_1,b_1),\dots,(a_N,b_N)$, and for each pair ...

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232 views

### Reference on (discrete) log-concave probability distributions

A discrete distribution $p$ over $\mathbb{N}$ is said to be log-concave if it satisfies the following conditions:
The support of $p$ is a contiguous interval, i.e. $\exists a \leq b$ s.t. $p_i > ...

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90 views

### Lower bound on the tail of the hypergeometric distribution

Suppose there is a bag with $M$ white marbles and $N - M$ black marbles. Let $H(n, N, M)$ be a random variable which is number of white marbles in a draw, without replacement, of $n$ marbles from a ...

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158 views

### Volume of randomly changing sphere follows beta distribution

We are given $X,X_1,\ldots,X_N$ independent and identically distributed $k$-dimensional vectors. For a given query point $X_q\in\mathbb{R}^k$ assume without loss of generality that $X_1,\ldots,X_m$ ...

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168 views

### Central limit theorem with degenerate covariance matrix

Are there known generalisations of the central limit theorem for several random variables when the covariance matrix is degenerate?
The usual proof of CLT based on characteristic functions (see e.g. ...

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### Shift invariance for the distribution of quadratic polynomials

For a probability distribution $X$, supported on integers, define the shift-invariance of $X$, denoted by $shift(X)$ = total variation distance between the random variable $X$ and $X+1$.
Let ...

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43 views

### Taking power of the integrand in a Riemann-Stieltjie Integral

This is a problem I am trying to solve as part of a calculation for Value-at-Risk.
Given that
$P(X<x)=F(x)=\int_{\theta}F(x|\theta)dG(\theta)=1-\alpha$,
where $F$ and $G$ are CDF's, is there a ...

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146 views

### Asymptotic behavior of $X_n$ in a Dirichlet vector $(X_1, …, X_n)$

Let $(\alpha_k)$ be a sequence of positive numbers and let $(Y_k)$ be a sequence of independent random variables $Y_k \sim \text{Gamma}(\alpha_k,1)$. Set $X_n=\dfrac{Y_n}{\sum_{i=1}^nY_i}$.
(edit) ...

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183 views

### Inequality with CDF of order statistics

here is a problem I have been struggling with for a while now. This is for a paper I am working on. Any help would be appreciated! Here we go:
Each bidder's valuation $\theta _{i},$ $i=1,...,N$, is ...

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280 views

### Weak convergence of random measures

Let $\mu_n,n\in \mathbb N$ be a random probability measures and let $\mu$ be a deterministic probability measure on $\mathbb R$. That is to say, that the $\mu_n$ are measurable maps from a probability ...

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### Rate of Convergence of Compound Poisson Laws to Infinitely Divisible Laws

It is known that every infinitely divisible random variable is the limit in law of a sequence of compound Poisson random variables (see for instance Theorem 1.2.18 of Lévy Processes and Stochastic ...

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51 views

### Numerical approximation to the Wasserstein metric?

Are there numerical methods for approximating/calculating the Wasserstein metric in particular cases?
Suppose that $f$ and $g$ are two density functions with the same support. How can I calculate the ...

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39 views

### Error for the convergence by distribution

A sequence of random variables $X_n$ converges in distribution to $X$, if there is pointwise convergence of its characteristic functions, i.e. $\lim_{n\rightarrow\infty}\phi_{X_n}(\lambda) = ...

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180 views

### An inequality based on expectation of continuous random variables

I am trying to prove the following statement:
$$
E[g(X)] E[X^2g(X)]\ge E[Xg(X)] E[Xg(X)]
$$
where $X$ is a random variable, $E[\cdot]$ denotes the expectation operator with respect to ...

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118 views

### Random partitions with prescribed pairwise membership probabilities

Let $(p_{ij}) \in [0,1]^{n \times n}$ be a given symmetric matrix, with $1$ on the diagonal. Suppose $\pi$ is a partition of $[n]=\{1,\dots,n\}$ and let us write $i \stackrel{\pi}{\sim} j$ if $i$ and ...

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162 views

### Convergence in the Wasserstein metric and the square root function

Let $f$ be a smooth probability distribution on the unit square $S$ such that $f(x)>0$ on $S$. Let $\{g_i\}$ be a sequence of smooth probability distributions such that $g_i(x)>0$ on $S$ as ...

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40 views

### A curious example envolving moment's convergence

Let $\{X_n\}$ be a random variable sequence and $X\sim N(0,\sigma)$. In general, the convergence $E(X_n^k) \stackrel{n}{\longrightarrow}E(X^k)$ doesn't implie that $E(X_n^{k+1}) ...

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41 views

### Sensitivity of inverse normal cdf

Let $Q^{-1}$ be the inverse function of a standard normal CDF. For $0 < \epsilon < p,p' < 1 - \epsilon$, how much does the function $Q^{-1}$ change as a function of $|p - p'|$? Any useful ...

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107 views

### 1-wasserstein distance v.s. total variation distance

Suppose that $\mu_1$ and $\mu_2$ are two distributions defined on $\mathbb{R}^n$ and $\gamma$ is a symmetric distribution (around $0$) on $\mathbb{R}^n$ with compact support. Let $\gamma_x$ denote the ...

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112 views

### Discrete Maximum Entropy Distribution with given mean

For a given mean $\mu$, what is the entropy maximizing probability distribution on the nonnegative integers?
Different sources indicated either the geometric or the Poisson distribution for this. As ...

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177 views

### Book on Convergence Concepts in Probability without Measure Theory [closed]

I am looking for a comprehensive book on Probability which discusses Convergence of Random Variables in detail, excluding portions of Measure Theory. Allan Gut's "Probability: A Graduate Course" seems ...

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94 views

### Generating random variables from the Cantor Distribution [closed]

I am looking for a method (exact, if possible, but at least asymptotically correct) for generating random variates from a Cantor Distribution? It seems like its abstract definition prevents this. In ...

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56 views

### convergence in distribution and convergence of moments

Suppose that the sequence of r.v $\{X_{n}\}_{n\geq 1}$ has all the moments, and $X_{n}\stackrel{D}{\longrightarrow}X\sim N(0,\sigma)$. Assume that $E\left\{(X_{n})^{K}\right\} ...

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60 views

### Sampling point uniformly at random satisfying equality constraints

First of all, I apologize in advance if the question has already been asked in some way on this site and/or if there is a widely known solution to this problem.
The description of my problem is ...

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94 views

### Do constrained random walks converge weakly to the Wiener measure on the space of constrained paths (that corresponds to the heat equation)?

Let $U$ be an open subset of $\mathbb{R}^n$ such that
$\partial \overline{U}$, the boundary of $\overline{U}$ is ''nice''
(for simplicity you can assume piecewise smooth). I also want to allow the ...

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161 views

### Probability question involving drawing balls from an urn

Suppose there's an urn containing $r$ red balls and $b$ blue balls. At each trial, I'm drawing a ball at random from the urn, without replacement. Let $R$ denote the event of drawing a red ball, and ...

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99 views

### Multiple Bipartite graphs and matchings

I've been told recently that it's better i just for help regarding my 'specific' problem rather than lots of little questions around the same topic which appear somewhat unclear. I would first like to ...

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### 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 $1$ ...