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.

Filter by
Sorted by
Tagged with
10 votes
4 answers
638 views

The min of the mean of iid exponential variables

Let $X_1, \ldots, X_n, \ldots$ be iid exponential random variables with mean 1. It is well-known that $\min_{1\le j < \infty} \frac{X_1 + \cdots + X_j}{j}$ follows the uniform distribution U(0,1). ...
John Wong's user avatar
  • 753
2 votes
0 answers
140 views

Adding elements in a list *in expectation*

Suppose $𝐿_1,…,𝐿_𝑘$ are lists with $n$ elements each. We use a fully independent hash function ℎ to compute a value for each element of each list. (We suppose the hash function returns a value ...
user1246462's user avatar
0 votes
1 answer
76 views

Distribution of line segment intersections in random pointsets

let $P$ be a set of $n$ points that are uniformly distributet inside the unit square ore unit circle, and $L=\lbrace\ell_{ij}\rbrace := \lbrace \lbrace \alpha p+ (1-\alpha q)\rbrace\,|\,0\le\alpha\le ...
Manfred Weis's user avatar
  • 12.6k
-1 votes
1 answer
48 views

Linear operator over a simplex space in a multinomial distribution parameter estimation problem

This is actually a variant of a well-known problem of how the parameters of a multinomial distribution can be estimated by maximum likelihood, and this arises from a final year project I undertook ...
Hephaes's user avatar
5 votes
2 answers
173 views

Integrability of Gaussian sums

Let $(X_1, \ldots, X_n)$ be a Gaussian vector, and $Z = \sum_{i=1}^n |X_i|$. Since the map $x \mapsto e^{x^2}$, is convex, for any $t>0$ $$ e^{tZ^2} \, = \, e^{t \big(\sum_{i=1}^n |X_i| \big)^2}...
Paul's user avatar
  • 51
3 votes
0 answers
455 views

From $f$-divergence to its dual: the transformation of convex functions on $\mathbb R_+$ by $f^*(t) = 1 f(\frac 1 t)$

I would like to understand the relationship between minimising the KL divergence $P \mapsto D_{KL}[P,Q]$ and the reverse KL divergence $P\mapsto D_{KL}^*[P,Q]=D_{KL}[Q,P]$ for probability measures $P$ ...
Lance's user avatar
  • 203
3 votes
1 answer
550 views

Random variables with no first moment

Consider a random variable $X$ with $\mathbb E(\vert X \vert)=\infty$. I am then wondering if this implies that for $X_i$ iid copies of $X$ $$\limsup_n \frac{\vert \sum_{i=1}^n X_i \vert}{n}=\infty?$$ ...
Doob's user avatar
  • 39
2 votes
2 answers
372 views

Probabilistic combinatorial optimization problem on the distances between pairs of points in $[0,1]$

Let $S$ be a set of $n \gg 1$ points lying on the interval $[0,1]$. Given a point $p\in[0,1]$, let $S_p\subseteq S\times S$ be the set formed by all pairs of points $(x,y)$ with $x,y\in S$, such that ...
Penelope Benenati's user avatar
0 votes
1 answer
121 views

Projection onto manifold of Gaussian measures by "trunction" of moments

Let $\mathcal{P}_2(\mathbb{R}^n)$ be the set of Borel probability measures on $\mathbb{R}^n$ with finite mean and variance; in the sense that $$ \int_{x \in \mathbb{R}^n} \|x\|^p d\mathbb{P}(x) < \...
ABIM's user avatar
  • 5,019
1 vote
0 answers
98 views

Cartesian product of Poisson processes

Consider $n$ smooth, compactly supported functions $\phi_1,\dots, \phi_n \in C_c^\infty(\mathbf{R})$, and generate $n$ independent Poisson spatial processes $N_1,\dots,N_n$ on $\mathbf{R}$, each with ...
Jacob Denson's user avatar
0 votes
1 answer
316 views

Expectation of the ratio of two discrete random variables with combinatorial constraints

We are given a set $S=\{1, 2, \ldots, n\}$ where $n\gg 1$, and for all indices $1\le i \le n$, $i$ is associated with a real value $\alpha_i\!\cdot\! v_i$, where $\alpha_i\in[0,1]$ and $v_i\in(0,1]$. ...
Penelope Benenati's user avatar
0 votes
1 answer
124 views

Question on limit in probability of the ratio of max to min of 2 sequences of non-ive, continuous iid random variables with support $[0, \infty).$

For each $ m \ge 1$, let $X_m$ and $Y_m$ be two non-negative iid random variables with the same distribution. (The distributions of $X_m$ may change with different $m$.) **Assume that their support of ...
Learning math's user avatar
22 votes
1 answer
5k views

Does Pinelis' inequality (1994) exist?

I am reading a paper on stochastic optimization. And in this paper, the proofs are based on the Pinelis' 1994 inequality. I read the paper by Pinelis for more information and it is with great ...
K.J Fogang Fokoa's user avatar
2 votes
1 answer
1k views

Finding the expectation $\mathrm{E} (1/ X)$ for a negative binomial random variable $X$

Suppose a random variable $X$ is distributed as $\operatorname{NB}(\mu, \theta)$, and its mass is as follows $$ \mathrm{P}(X = y) = \binom{y + \theta - 1}{y} \left(\frac{\mu}{\mu + \theta}\right)^{y}\...
香结丁's user avatar
  • 331
3 votes
0 answers
172 views

Probability terminology

This is strictly a low-level terminology question. If I have a probability space $\Omega$ and a measurable space $S$, then a random variable $X:\Omega\rightarrow S$ gives rise via pushforward to a ...
Steven Landsburg's user avatar
0 votes
2 answers
343 views

If $\mu$ is an infinitely divisible probability measure on $[0,\infty)$, then the Lévy measure of $\mu$ is the vague limit of $n\mu^{*1/n}$

If $\nu$ is a finite measure on $(\mathbb R,\mathcal B(\mathbb R))$, let $\nu^{\ast k}$ denote the $k$-fold convolution¹ of $\nu$ with itself for $k\in\mathbb N_0$, $$\exp(\nu)\mathrel{:=}\sum_{k=0}^\...
0xbadf00d's user avatar
  • 161
0 votes
0 answers
141 views

Total variation convergence of random matrices and convergence of empirical spectral distributions

In the paper https://arxiv.org/pdf/1411.5713.pdf, on page 17, the authors prove in Theorem 7 that the total variation distance between the joint distribution of the entries of certain Wishart matrices ...
Carbon's user avatar
  • 1
2 votes
0 answers
49 views

What are some beginner's references on algebraically structured (statistical) models, and their connection with group actions and Fourier transform?

I asked this question on Cross Validated a few days ago, but didn't really get a favorable response, so asking here to see if I get any. I'm looking at the description of a short-term position in ...
Stat_math's user avatar
  • 223
1 vote
1 answer
151 views

If $L_t=\sum_{i=1}^{N_t}Y_i$ is a compound Poisson process, then $\left|\left\{s\in[0,t]:\Delta L_s\in B\right\}\right|=\sum_{i=1}^{N_t}1_B(Y_i)$

Let $H$ be a $\mathbb R$-Hilbert space, $\mu$ be a finite measure on $\mathcal B(H)$ with $\mu(\{0\})=0$ and $(L_t)_{t\ge0}$ be a $H$-valued càdlàg Lévy process on a probability space $(\Omega,\...
0xbadf00d's user avatar
  • 161
3 votes
1 answer
118 views

The mean value of the reconstruction complexity of a random sequence

This problem is motivated by the problem of reconstructing a genome from the family of its short subwords. Given a word $w$ and a positive integer $k$, let $M_k(w)$ be the family of all subwords of ...
Taras Banakh's user avatar
  • 40.8k
1 vote
1 answer
256 views

Decomposition of the sum of nonnegative random variables [closed]

Non-necessarily independent random variables $X_1,~X_2,~\cdots,~X_n$ are supported on $[0,a_1],~[0,a_2],~\cdots,[0,a_n]$ and with mean values $\mu_1,~\cdots,~\mu_n$ respectively, where all $a_i$ and $\...
RyanChan's user avatar
  • 550
2 votes
1 answer
203 views

sign of odd central moments of binomial distribution

I am interested in the sign of odd, central moments of a binomial distribution. From DOI. 10.1137/070700024 I have the formulae: $ E\left[\left(X-\mu\right)^d\right]= \sum_{i=0}^n\binom{n}{i}\left(-p\...
qwert's user avatar
  • 89
7 votes
2 answers
256 views

An inequality involving the beta distribution

Let $F$ be the CDF for a Beta distribution, $$F(x)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\int_{0}^{x}t^{a-1}(1-t)^{b-1}\,dt$$ with $a,b\geq 1$. Is it true that $$\frac{b}{a+b} \leq\int_0^1\sqrt{F(x)}...
Tom Solberg's user avatar
  • 3,929
8 votes
1 answer
2k views

General Fourier inversion formula (Gil-Pelaez)

Gil-Pelaez (1951) proves the Fourier inversion formula \begin{align*} F(x) &= \frac{1}{2} + \frac{1}{2\pi} \int_0^\infty \frac{e^{itx}\phi(-t)-e^{-itx}\phi(t)}{it}dt \\ &= \frac{1}{2} - \frac{...
Alex's user avatar
  • 255
1 vote
1 answer
305 views

Posterior expected value for squared Fourier coefficients of random Boolean function

Let $f : \{0, 1\}^{n} \rightarrow \{-1, 1\}$ be a Boolean function. Let the Fourier coefficients of this function be given by $$ \hat f(z) = \frac{1}{2^{n}} \sum_{x \in \{0, 1\}^{n}} f(x)(-1)^{x \cdot ...
RandomMatrices's user avatar
4 votes
0 answers
146 views

Can we show equivalence of two distributions based on their statistics?

Let $p,q$ be two distributions on $\mathbb{R}^d$. Let $f:\mathbb{R}^d\times\mathbb{R}^d\rightarrow\mathbb{R}$. Under what conditions does $\mathbb{E}_{x\sim p}f(x,z)=\mathbb{E}_{x\sim q}f(x,z)\ \...
Zhifeng Kong's user avatar
1 vote
1 answer
306 views

How can one calculate distribution of ratio of differences of order statistics of uniform distribution?

Let $X_1, ..., X_n \quad i.i.d \sim U[a,b]$ Then $Z_i$ defined as: $$ Z_i = \frac{X_{(i)}- X_{(1)}}{X_{(n)} - X_{(1)}}, \quad i = \overline{2,n-1}, $$ where $X_{(k)}$ is the $k$-th order statistic. I ...
Nourless's user avatar
  • 145
0 votes
1 answer
419 views

Laplace transform inversion

I have a probability distribution that is defined through it's Laplace transform by : $$L(t) = \mathbb E(e^{-tX}) = e^{1 - \frac{1+t}{t}\ln(1+t)}$$ Using R and the invLT package, i have a numerical ...
lrnv's user avatar
  • 686
2 votes
1 answer
249 views

A distribution such that these expectation are 'closed-form'

I am seeking a continuous distribution with real positive support for the random variable $X$ such that, for all $t \in \mathbb R_{+}$, $$\mathbb E \left(\ln\left(1+tX\right)\right)$$ is given in a '...
lrnv's user avatar
  • 686
2 votes
0 answers
83 views

approximate the square of 2-norm distance between binary distributions with high probability

Suppsose we take $m$ samples from a Bernoulli distribution with probability $p$, and $m$ samples from another probability distribution with probability $q$. We want to calculate a statistic $x$ from ...
gondolf's user avatar
  • 1,487
1 vote
0 answers
145 views

Is there a name for a random variable that is the absolute value of the difference between two iid discrete uniform variables?

I'm working on a project and I needed to calculate the distribution of the difference between two iid discrete uniform variables (sorry for the long title). That is, let $I, J$ be two iid discrete ...
Simon's user avatar
  • 27
3 votes
1 answer
394 views

Symmetric distribution optimization problem of distances between points in $[0,1]$

Let $\mathcal{D}$ be a probability distribution with support $[0,1]$. Let $x, y, z$ the outcomes of three i.i.d. random variables $X, Y, Z$ with distribution $\mathcal{D}$, sorted in increasing order, ...
Penelope Benenati's user avatar
0 votes
0 answers
81 views

Linear independence of Wishart matrices

Let $W\sim W_n(I,d)$ be a real Wishart matrix of an identity covariance matrix and $d$ degrees of freedom, i.e., $W=XX^T$ for $X$ being an $n\times d$ matrix whose entries are i.i.d sampled from a ...
user50394's user avatar
  • 123
2 votes
1 answer
216 views

Probability distribution optimization problem of distances between points in $[0,1]$

Let $\mathcal{D}$ be a probability distribution with support $[0,1]$. Let $X, Y, Z$ three i.i.d. random variables with distribution $\mathcal{D}$, and $T$ a random variable uniformly distributed in $[...
Penelope Benenati's user avatar
1 vote
1 answer
297 views

Dual problem with integrals

I am reading a paper where the author derives the following Lagrangian dual problem : $\min_v \int_R \frac{1}{4} \frac{\beta^2}{v-2\|x\|}dx+v\;\;\;\text{s.t.}\;\;\;v\geq 2\|x\|\;\;\;\forall x \in R$ ...
OmarR's user avatar
  • 67
0 votes
0 answers
159 views

The reason why a test is undersized?

Now I have a statistic $T_n$ for testing $H_0 \leftrightarrow H_1$, and I have proved that: $$n T_n \rightarrow_d \chi_K^2$$ under $H_0$. Then an asymptotic $\chi^2$ test can be used, an asymptotic ...
香结丁's user avatar
  • 331
0 votes
0 answers
36 views

How to derive the asymptotic distribution under the alternative hypothesis?

In the class, my professor introduced the ADF test, and I suddenly realized that it seems that all tests are under the null hypothesis. I'm curious that if it is possible to make the judgement under ...
ZhiJie Fu's user avatar
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
1 vote
1 answer
220 views

Scaling behavior of Wasserstein distances

Let $p>1$ and $\mu\neq \nu$ be two probability measures on $\Omega\subset \mathbb{R}^d$ a bounded set. For $\alpha \geq 0$, we let $$C_\alpha(\mu,\nu) = \inf_\sigma \frac{W_p(\mu+\sigma,\nu+\sigma)}...
Vincent's user avatar
  • 83
1 vote
0 answers
44 views

How to use the mixed normal distribution to construct a proper statistics?

For a random vector $\xi_n \in \mathbb{R}^p$, if $\xi_n \rightarrow_d N(\mu, \Sigma)$, we can construct \begin{equation*} \Psi := \xi_n^{\top} \widehat{\Sigma}^{-1} \xi_n \end{equation*} for ...
香结丁's user avatar
  • 331
1 vote
1 answer
1k views

Convolution of two Gaussian mixture model

Suppose I have two independent random variables $X$, $Y$, each modeled by the Gaussian mixture model (GMM). That is, $$ f(x)=\sum _{k=1}^K \pi _k \mathcal{N}\left(x|\mu _k,\sigma _k\right) $$ $$ g(y)=\...
wuhanichina's user avatar
0 votes
1 answer
302 views

Anomaly with a pdf

The pdf of the range $\omega$ of $n$ identically r.v.'s random variables distributed with cdf $\mathbf{F}$ and pdf $\mathbf{f}$ is given by $$ g(\omega)=n(n-1) \int_{-\infty}^{\infty} f(x)[F(...
AgnostMystic's user avatar
2 votes
1 answer
730 views

Diagonalizability of Gaussian random matrices

Let $X$ be an $n\times n$ matrix whose elements are i.i.d. sampled from a normal distribution of zero mean and unit variance. Is $X$ diagonalizable over $\mathbb{C}$ with probability 1? Is there a ...
user50394's user avatar
  • 123
1 vote
0 answers
32 views

Condtions for a stochastic process to be locally non-factorizable

Given a stochastic process $X=(X_t)_{t\in I}$ on $\mathbb{R}^d$ with continuous sample paths supported on a prob. space $(\Omega, \mathscr{F}, \mathbb{P})$ and such that each pair $(X_s, X_t)$, with $(...
fsp-b's user avatar
  • 421
0 votes
1 answer
48 views

Emergence of non-power-law behaviour under infinite summing

Suppose $X_1,X_2,...$ is a sequence of random vectors in $\mathbb{R}^n$ s.t for all $k \in \mathbb{Z}^+$ and $u \in \mathbb{R}^n$ we have that $E [ \langle u, X_i \rangle ^k]$ is finite. (The $X_i$s ...
gradstudent's user avatar
  • 2,136
2 votes
0 answers
227 views

An inequality of KL Divergence for two different distributions passing through a same channel

Let $X$ be a random variable which takes values in $\mathcal{X}$. Assume that we pass $X$ through two independent conditional pdf $p_{X_1|X}$ and $p_{X_2|X}$ and choose $X_1$ with probability $\lambda$...
Math_Y's user avatar
  • 311
1 vote
2 answers
110 views

If a joint density factorizes on a square, does this imply that the marginal random variables are locally independent?

Let $Z=(X,Y) : \Omega\rightarrow\mathbb{R}^2$ be a Borel-measurable random vector and $U\subset\mathbb{R}$ be open. Suppose that $Z$ is absolutely continuous with Lebesgue density $\zeta$. I was ...
fsp-b's user avatar
  • 421
1 vote
1 answer
127 views

How much reduction in expected variance can we get from a single bit?

Consider the following protocol: Alice has a number $X$, chosen according to a known distribution $\mathcal D$ (e.g., $X\sim U[0,1]$). She can send a bit to Bob, giving him more information about $X$ (...
M A's user avatar
  • 127
2 votes
1 answer
146 views

A limit question involving Cramer's decomposition of normal random variables

I've come across the following question. Say we have two families of random variables, $X_N$ and $Y_N$, such that $\mathbb{E} X_N=\mathbb{E} Y_N=0$ and $\mathbb{E}X_N^2=1$. Now assume that: $$|\mathbb{...
Pax's user avatar
  • 821
0 votes
1 answer
546 views

Concentration of $\ell_2$ norm of a vector sampled from a distribution

Let $X=(X_1,\ldots,X_n)$, where $X_i \sim P_{p_i}(0,\frac{1}{\lambda})$ are iid, $P_{p_i}$ is sub gaussian distribution for $i^\text{th}$ element, and 0 and $1/\lambda$ are mean and variance. I'm ...
newbie's user avatar
  • 61

1
11 12
13
14 15
39