All Questions
Tagged with pr.probability probability-distributions
1,384 questions
3
votes
0
answers
50
views
A transformation of a probabilistic distribution
Let $\mu$ be a probabilistic distribution on $\mathbb{N}$.
Consider the following random process $F_i(\mu)$.
First we choose $i$ numbers $x_1, \ldots, x_i$ randomly with respect to distribution $\...
- 2,576
3
votes
0
answers
841
views
Is uniform distribution on unit sphere subgaussian?
Is uniform distribution on unit sphere subgaussian?
To be specific, let $X = (X_1,\dots,X_d) \sim \mbox{Unif}(\mathcal{S}^{d-1})$. What is the Orlicz-$\psi_2$ norm of $X$?
- 719
4
votes
0
answers
103
views
Does a non-exchangeable empirical reverse-martingale exist?
Consider a possible finite sequence $\xi_1,\xi_2,\dots$ of random variables and consider the measure-valued empirical process
$$\eta_n=\frac{\sum_{i=1}^n\delta_{\xi_i}}{n},\:\:\: n=1,2,\dots$$
Assume $...
- 211
3
votes
1
answer
345
views
Second moment of cos(x,y) for Normal x,y?
I'm trying to figure out second moment of the following quantity
$$y = \frac{\langle x_1, x_2 \rangle}{\left\|x_1\right\|\left\|x_2\right\|}$$
Where $x_1$, $x_2$ are sampled independently from $\...
- 2,242
0
votes
1
answer
171
views
Distance of distributions of random variables, without PDF
Consider an interval $I$ with a smooth probability measure $d\mu (x) = c(x) dx$ and two known real measurable functions $f_1(x)$,$f_2(x)$. Both functions define a distribution on $X = {\rm Im} \, [f_1]...
- 3,574
4
votes
2
answers
475
views
midpoint between two normal distributions for the Rao-Fisher metric
Given two multivariate gaussian distributions $G_0 \sim N(\mu_0,\Omega_0)$ and $G_1 \sim N(\mu_1,\Omega_1)$, is there a closed-form formula for the gaussian distribution equidistant from them that is ...
- 427
2
votes
0
answers
107
views
Markov chain approximates a fractional diffusion
Let assume that
$$
dX_t=\mu(X_t)dt+\sigma(X_t)dW_t^H, X_0\in \mathbb{R}
$$
Where $\mu(.), \sigma(.)$ satisfy some conditions that guarantee $X_t$ exists, and $dW_t^H$ is a fractional Brownian motion ...
- 323
1
vote
0
answers
809
views
Generalized Chi squared distribution
What is the distribution of $Y Y^\top$ if $Y \sim N(\mu,\Sigma)$ is a generic multivariate gaussian vector?
- 719
7
votes
3
answers
346
views
Concentration Bound of $0/1$ permanent
If I pick a random $0/1$ $n\times n$ matrix with $0$ occuring with probability $p$ then what does the distribution of the permanent look like?
- 13.9k
2
votes
1
answer
233
views
Difference of hypoexponential distributions
Suppose that we have two random variables defined on the same sample space $\Omega$
$X\sim \text{Hypoexp}(\alpha_1,\dots,\alpha_n)$ and $Y\sim \text{Hypoexp}(\beta_1,\dots,\beta_m)$
or, equivalently,...
- 21
0
votes
1
answer
201
views
The distribution of the maximum of a series of extreme value type I random variable
I have an infinite series of independent identically distributed random variables $\{X_i\}_{i=1}^\infty$ which follows extreme value type I distribution which can be found [here] (https://en.wikipedia....
- 35
2
votes
0
answers
156
views
Sufficient condition for a solution to Hamburger moment problem
Let $\{m_n\}_{n=0}^{\infty}$ be a sequence of real numbers.
It is well known that there exist a positive Borel measure $\mu$ on the real line with moments given by $\{m_n\}_{n=0}^{\infty}$ if and ...
- 671
2
votes
0
answers
171
views
Distribution for the extreme values of a cumulative sum of normal variables
Suppose I have a sample $X$ of $n$ iid Normal random variables $(X_1,X_2,..,X_n)$.
Now, define the sample mean $u=\frac1n\sum_{i=1}^n X_i$ and let $Y_i=X_i-u$.
Let $Z_k=\sum_{i=1}^k {Y_i}$. Note ...
- 121
0
votes
0
answers
141
views
Effect of partitioning the realizations of random variables on the total variation distance?
Let $X$ and $Y$ be two random variables with joint pmf $p(x,y)=p(x)\cdot p(y|x)$ and $X$ has uniform distribution. Also assume that the following relation is satisfied:
\begin{align}
\lVert p(y|x)-p(y)...
- 287
2
votes
1
answer
352
views
Density of Non-Homogeneous Poisson Process
Given $\lbrace Y_i\rbrace$ a non-homogenous Poisson process with mean density $\theta y^{-1}e^{-y}$ where $y>0$ $(\theta>0)$. I.e., the number of points of $\lbrace Y_i\rbrace$ in $(a,b)$ with $...
- 329
1
vote
1
answer
124
views
Reconstructing the number of distinct elements from a random projection
Assume we have an unknown sequence $x_1,\ldots, x_n\in \mathcal U$.
We get to observe the sequence $h(x_1),h(x_2),\ldots, h(x_n)$, where $h:\mathcal U\to \{1,\ldots, k\}$ is a random function such ...
- 618
0
votes
1
answer
80
views
Expectation of ratio between product of gaussian r.v.'s and generalized gamma r.v
Given
\begin{equation}\label{eq:definition_of_z}
\begin{split}
\textbf{Z} = \left[\begin{array}{cccc}
{z}_{11} & {z}_{12} & \cdots & {z}_{1P} \\
{z}_{21} & {z}_{22} & \cdots & {...
2
votes
1
answer
71
views
Distances between probability distributions by the variance of the test functions
Let $P$ and $Q$ be two probability distributions on $\mathbb{R}$. The goal is to obtain a notion of ``distance'' between $P$ and $Q$, e.g., total variation distance, K-L divergence.
Let $f\colon \...
- 1,127
0
votes
0
answers
114
views
Merging Poisson/lognormal processes
We know that merging two Poisson processes results in another Poisson process with a rate that is the sum of the two original rates.
(https://www.probabilitycourse.com/chapter11/...
4
votes
2
answers
1k
views
Total progeny of a Galton-Watson branching process - standard textbook question
While analyzing some parallel-computing related algorithm, I came across a probability distribution with a particularly nice property (at least to me), but I am unable to write it down explicitly.
...
- 396
4
votes
0
answers
162
views
Concentration Inequality for Score Functions of Exponential Familty
Let $p$ be the density of a continuous one-parameter exponential family distribution on $\mathbb{R}$. We assume that
$$p(x) = c(x)\cdot \exp\bigl [ x \cdot \theta - b(\theta ) \bigr ], $$
where $\...
- 1,127
3
votes
0
answers
82
views
For a given Gaussian vector, which rectangular parallelepiped with unit volum has the largest probability?
Let $X$ be a centered Gaussian vector of $\mathbb{R}^n$ and $\Gamma$ its covariance matrix. We assume that diagonal coefficients of $\Gamma$ are all equal to 1.
We are looking for a rectangular ...
- 103
1
vote
0
answers
49
views
A question about the prediction error
I am reading about the prediction error estimation and I found the following:
Suppose we have ${\mathbf{Y}}=\mathbf{x}_0+ \epsilon$, where, $\epsilon$ is normally distributed as $\sim \mathcal{N}(0, \...
- 11
0
votes
0
answers
57
views
Parametric distribution where the parameter follows a diffusion process
I'm looking for a distribution $P_{\theta}$ with pdf $f (t,\theta)$ over $\mathbb{R}^{+}$ such that there exists functions $\mu(\theta)$ and $\sigma(\theta)$ such that for all $t>0$:
$$\mu(\theta)\...
- 1,902
3
votes
3
answers
292
views
A question in central limit theorem
Suppose $\{X_n,n\ge1\}$ are independent r.v., $E(X_n)=0$, $\operatorname{Var} \left(X_n\right)=\sigma_n^2<\infty$. Set $S_n=\sum_{i=1}^nX_i$ and $s_n^2=\sum_{i=1}^n\sigma_i^2$, assume
$$\frac{S_n}{...
- 141
0
votes
1
answer
151
views
Can an unskewed distribution be expressed as product of a normal and another distribution?
Let $x$ be a continuous random variable with zero mean and zero skew. What are the conditions under which we can say that $x$ can be expressed as the product $z y$ where $z$ is a standard normal and $...
- 620
3
votes
1
answer
190
views
Solution for Moment problem
I want to invert a sequence of moments and find a function f(x) satisfying:
$$m_r=\int x^{r}f(x) dx=\int x^{r} dF(x)$$
The sequence of moments is given by:
$m_{2s+1}=0$
$m_{2s}=\sum_{k=1}^{s}\binom{...
- 333
6
votes
1
answer
774
views
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 $f$ ...
- 519
10
votes
1
answer
263
views
q-versions of the geometric distribution and their names
I'm trying to set straight various $q$-deformations of the standard geometric distribution.
The geometric distribution on $\left\{ 0,1,\ldots \right\}$ is well-known, it has
$$
\mu_1(X=j)=(1-p)p^j,\...
- 1,765
1
vote
0
answers
575
views
Bounding the total variation distance of two specific random variables?
Let $X$ and $Y$ be two independent discrete random variables, and $Z$ be a function of $X$ and $Y$, i.e., $Z=f(X,Y)$. Suppose that $\Gamma$ is a set such that
$$\mathrm{Pr}[(X,Y)\in\Gamma]\geq 1-\...
- 287
1
vote
0
answers
43
views
Distribution of maximum minor of a random matrix with one special column
Given $m,n,\ell\in\Bbb N$ and $\beta\in(0,1)$ consider the uniformly picked random matrix $A\in\Bbb Z^{n\times (n+1)}$ with $0\neq|\mathsf{det}(A^\circ)|\leq m^{\frac 1\ell}$ where $A^\circ$ is the ...
- 13.9k
3
votes
0
answers
75
views
Covariance of censored/clipped Gaussians
I am interested in the covariance of two clipped (or censored) Gaussian variables.
More precisely, let $g_1 \sim N(0,\sigma_1^2)$ and $g_2 \sim N(0,\sigma_2^2)$ be two (dependent) Gaussians with $\...
- 53
3
votes
0
answers
303
views
Exchangeable or iid random variables and linear conditioning
Let $X_1,\ldots ,X_N$ be independent identically distributed random variables (or, more generally, exchangeable random variables,
but let's assume independence for simplicity). Then
$$
E(X_i\mid X_1+\...
- 1,765
2
votes
0
answers
103
views
measures in infinite dimension space of entire functions [closed]
It is known that there is no canonical generalization of Lebesgue measure in infinite dimension of function spaces. Since it seems that the space of (transcendental) entire function seems improtant ...
- 1,706
1
vote
0
answers
110
views
Approximating or calculating the mutual information of certain binary random vectors
In my studies of applied probability I have recently met the following problem on which I need help:
We consider two binary random (column) vectors $ X,Y \in \{0,1\}^d $ where the mutual ...
- 620
4
votes
1
answer
141
views
Fuzzy layers in graphs and neural networks
I wonder if the following statistical description of the layer architecture of finite graphs has been considered before and where I can find some references (especially under which name).
Consider a ...
- 9,720
3
votes
2
answers
1k
views
Is there a notion of Convergence in PDF/PMF
I am learning about local limit theorems. The following example is probably why we don't have a "convergence in density/pmf."
Ex: $X_1,X_2,\ldots$ is a sequence of independent RVs with mean $a$ and ...
- 329
1
vote
0
answers
58
views
A possible extension of the information bottleneck principle with added equality constraint on the conditional probability
This question is related to research on Tishby's information bottleneck principle as seen here, the problem at hand is inherently an optimization problem as seen directly on page 7 section 3.2, my ...
- 620
3
votes
1
answer
241
views
Rates of convergence for empirical quantization error
I'm looking for error rates of convergence for approximating a probability measure $P$ by a discrete probability with at most $k$ supporting points.
The setup I'm looking at is the following. Let $X$...
- 155
0
votes
1
answer
905
views
Parameter estimation distribution for hypergeometric distribution
Let the hypergeometric distribution is given by $h(k\mid N;M;n)$, where
$k$ is the number of observed successes,
$N$ is the population size,
$M$ is the number of success states in the population and
$...
- 749
1
vote
0
answers
77
views
Random variables with values in binary operations or in topologies of a certain set $X$
I wonder if the following situations have already been considered by mathematicians :
Random variables with values in a set of binary operations endowed
with a certain topology (or just with a $\...
- 129
4
votes
0
answers
95
views
Approximating martingales given marginal distributions
Let $(\mu_0,\mu_1)$ be a vector of probability measures on $\mathbb R$ that are of finite first moment, i.e.
$$\int_{\mathbb{R}}|x|\mu_i(dx)~<~+\infty \mbox{ for } i=0,1$$
and increasing in ...
- 1,835
2
votes
1
answer
139
views
Stochastic inverse
Let $X_t$ be a semi-martingale and $H_t$ be a predictable process and $g$ be a measurable bijective function with measurable inverse. Does there exist a function $f(h,x)$ satisfying
$$
\int_0^Tf(H_t,...
- 5,405
2
votes
0
answers
63
views
Sensitivity of a function against its random arguments
Let $g:R^{n+m} \to R$ be a deterministic function of some independent random variables $x_1,\ldots,x_n$ with distributions $f_{x_1}(x),\ldots,f_{x_n}(x)$ and some deterministic variables $z_1,\ldots,...
- 482
4
votes
1
answer
463
views
Variance and expectation of timed-change squared Bessel process
Let $X_t$ be a squared Bessel process satisfying the SDE:
$$
dX_t=\left(1-\frac{\beta}{(1-\beta)(1-\rho^2)} \right) dt +2\sqrt{X_t}dW^{(1)}_t
$$
and $v_t=v_0e^{-\alpha^2 t/2+\alpha W^{(2)}_t}$ be a ...
- 323
2
votes
0
answers
80
views
Maximal and second max of chi-squared (normal squared) distribution
Suppose $X_i$ are i.i.d.~$\log \chi^2$ where $\chi\sim N(0,1)$ distribution, in which case
$$
F(x) = \mathbb{P}\left( \log \chi^2 \le x \right)
= 2\Phi(e^{x/2}) - 1,
$$
and
$$
f(x) = e^{x/2}\phi(e^{x/...
- 719
4
votes
2
answers
3k
views
Entropy of the multinomial distribution
What is the entropy of the multinomial distribution? To fix notation, let us define $n > 0$ as the number of trials, $p_1, \ldots, p_k$ as the probabilities of each of the $k$ possible outcomes and ...
- 151
2
votes
1
answer
295
views
The asymptotics of $\int_{-\infty}^{\infty} \phi(x) {\Phi(\frac{x}{a})}^{qa} dx $ for normal distribution using saddle point approximation
In my probability and numerical analysis research I have come across the following predicament:
If we have a standard normal random variable X with CDF $ \Phi $, and PDF $ \phi $ I am interested in ...
- 620
0
votes
1
answer
503
views
Asymptotics of a 1D integral, or the orthant probability of an equicorrelated random Gaussian vector
Problem: Let $\phi(x)$ be the normal probability density function (pdf), and $\Phi(x)$ the normal cumulative distribution (cdf). I'm interested in the asymptotic behavior of the following integral
$I(...
- 968
-1
votes
2
answers
512
views
Deriving the joint distribution of multivariate normal transformed into Bernoulli
Given a covariance matrix $\sum_{ij}$ and a mean vector $\mu$ I have sampled $N$ multivariate normal vectors $Z = (z_1,...z_n)$ My goal is to create a vector of Bernoulli random variables $Y = (y_1,......
