The probability-distributions tag has no wiki summary.
3
votes
2answers
53 views
expectation of log(x+a) when X follows a beta distribution
Is there a closed form expression for the expectation of $\log(x+a)$ (with $a>0$, the case $a=0$ is obvious) when X follows a beta distribution?
6
votes
1answer
345 views
Mean of i.i.d Random Variables With No Expected Value
Let $X$ be an integer-valued random variable and let $X_n$ be the sum of $n$ independent realizations of $X$. I would like to understand the behavior of $X_n/n$ for large $n$ in some cases where $X$ ...
3
votes
1answer
82 views
Variance of maximum of mixture of gaussians
Let $\{X_i\}$ be an iid collection of standard normal $(N(0,1))$ random variables . Let $X = (X_1,\ldots,X_n)$, and consider a function of the form $f(X) = \max(A\cdot X)$, where $A$ is some ...
3
votes
0answers
41 views
Cramér-Wold theorem with independence assumption
Let $X = (X_1, X_2)$ be a random vector with joint probability density $p$. The celebrated Cramér-Wold theorem says that we can reconstruct $p$ from knowing the push-forward densities of $X$ under all ...
1
vote
1answer
43 views
Is there a simple closed form solution for the joint density distribution of an exponential distribution with a rate given by a Gamma distribution?
I have an exponential distribution with rate $\lambda$, where $\lambda$ is drawn from a Gamma distribution with shape and scale parameters $(k,\theta)$. I'd like to calculate an exact PDF for values, ...
5
votes
1answer
163 views
1-Wasserstein distance between two multivariate normal
The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by
...
1
vote
0answers
30 views
Does this kind of integral equations have unique solution?
Suppose $f_1$ and $f_2$ are two probability density functions on support $[0,1]$ (i.e. $f_1(x)=f_2(x)=0$ for any $x\not\in[0,1]$). Let $\varphi(x)$ denote a known probability density function on ...
-3
votes
0answers
14 views
Moment of uniform distribution [migrated]
Suppose that $U$ is a random variable from a uniform distribution on $[a, b]$.
Then, we can obtain the moment generating function of $U$, and by using that, we can get the $n$th order moment of $U$ ...
1
vote
0answers
47 views
Distribution of the local time for reflected Brownian motion on the quadrant
If $W$ and $B$ are two independent Brownian motions, $X_t=W_t-\sup_{u\le t} B_u + 1$. I want to find the distribution of $S$ the first hitting time of $0$ by $X$. If someone could give me a direction ...
5
votes
1answer
152 views
Convergence rate of the central limit theorem near the center of the distribution
I'm looking for fast convergence rates for the central limit theorem - when we are not near the tails of the distribution.
Specifically, from the general convergence rates stated in the Berry–Esseen ...
1
vote
1answer
63 views
explicit expressions of the distribution of sums of i.i.d. logistic random variables
Where can I find the explicit expression of the distribution of the sum of n i.i.d. logistic random variables, for n=2,3,4...
The expressions given in "On the convolution of logistic random ...
1
vote
0answers
48 views
What is entropy of a variable described by Knightian uncertainty? [closed]
I have asked this question at Theoretical Computer Science and received no response.
Given a discrete variable whose value is characterized by Knightian uncertainty, that is, belief and plausibility, ...
2
votes
3answers
144 views
Expected value of swaps
Suppose you have a list of non negative numbers of size N. Now you calculate the maximum element in the list by scanning the list linearly and constantly updating a variable which has initial value of ...
-1
votes
0answers
14 views
Transformation of a Random Variable heavyside step as transfer function [migrated]
I have a random variable X with probability distribution function of $P_{X}(x)= (1-e^{-2x})u(x)$, where $u(x)$ is the unit step function with $u(0)=1$.
I now have another random variable Y, which is ...
4
votes
3answers
323 views
Are there known expressions for total variation distance between $N(0,\sigma_1^2)$ and $N(0,\sigma^2)$
Are known expressions for total variation distance between $N(0,\sigma^2)$ and $N(0,\sigma^2+\epsilon)$ for small $\epsilon$? The only thing I seem to find is things are expression about the mean but ...
0
votes
0answers
66 views
Fitting distribution to spatial data
I am studying a physical process generating data which projects nicely into two dimensions with non-negative values. Each process has a (projected) track of $x$-$y$ points -- see the image below.
...
2
votes
0answers
45 views
Can truncated/non-smooth distributions be used as priors/posteriors in Variational Bayesian methods?
Variational Bayesian methods can sometimes be a good alternative to Markov Chain Monte Carlo numerical evaluation of probability distributions. They do this, as I understand it, by approximating the ...
1
vote
0answers
98 views
approximation of probability distribution
I have a question: Let $\mu$ be a probability distribution defined on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ satisfying
$$\int_{\mathbb{R}}|x|d\mu<+\infty$$
Set
$$A_n=\Big\{\frac{i}{n}:~ ...
1
vote
1answer
54 views
Euclidian norm of Gaussian vectors
Let $X \sim \mathcal{N}(0, \Sigma)$ be a Gaussian vector in dimension $N$. I am interested by the probability density of the random variable $\lVert X \lVert_2$.
If $\Sigma = {I}_N$, we recognize ...
1
vote
0answers
40 views
Cramér-Wold like theorem for independent random variables
Let $X$ be a random vector in $\mathbb R^n$ with probability distribution $\mathbb P_X$. Now when given only the family of distributions
\begin{align*}
\left\{ \mathbb P_{v_1 X_1 + \dots + v_n ...
4
votes
1answer
97 views
Does second order stochastical domination with increasing likelihood ratio imply first order domination?
This question is coming from the fact that all the counter examples for which second order stochastical domination holds but first oder stochastical domination fails do not accept increasing ...
4
votes
2answers
218 views
Estimate on gaussian distribution
Let X be an $\mathbb R^d$-valued random variable with distribution $N_d(0,\Sigma)$. I'm looking for a function $f$ such that
$$P(|X_1|\leq M, |X_2|\leq M,\dots, |X_d|\leq M)\geq f(M),$$
and such that ...
1
vote
0answers
39 views
Characteristic function known on subsets
Let $X$ be a random variable in $\mathbb R^n$ with distribution $\mathbb P_X$. Given a (infinite) family of matrices $W_t \in \mathbb R^{n \times m}$ parameterized by $t \in \mathbb R$, suppose we ...
3
votes
1answer
61 views
Relaxing conditions for Cramer-Wold type theorem
Let $X$ be a random vector in $\mathbb R^n$ with probability distribution $\mathbb P_X$. Now when given only the family of distributions
\begin{align*}
\left\{ \mathbb P_{v_1 X_1 + \dots + v_n ...
4
votes
1answer
169 views
Population dynamics for fish arriving via a Poisson process and living for a time given by some (not necessarily symmetric) general distribution
Imagine we have a hypothetical population of fish in a pond. The fish cannot reproduce, but are introduced by a Poisson process (with some known and fixed rate parameter independent of the total ...
0
votes
0answers
14 views
Distribution of the square of a AR process
Suppose given a discrete time AR process $\{h_k\}$ with the following dynamics:
$h_k = (1 + a)h_{k-1}-b+v_{k-1}$,
where $a, b$ are constants and $v_k\sim \mathcal{CN}(0,1)$ is complex gaussian ...
2
votes
1answer
84 views
Question about infinite-dimensional BM
Suppose we are given an $L^2(\mathcal{D})$-valued Brownian motion $W_t$ defined by
$$W_t:=\sum_{k=1}^{\infty}\sqrt{\sigma_k}W_t^k\phi_k(x),$$
where $\mathcal{D}$ is bounded domain in $\mathbb{R}^d$, ...
5
votes
1answer
157 views
“Smallest” event such that probability greater than a given value
Very briefly, consider the probability space $(\mathbb R^n, \mathcal{B}(\mathbb R^n),P)$. During a problem I am studying, I came to a point where i need to compute
\begin{equation*}
\begin{aligned}
...
5
votes
3answers
213 views
How do you call the problem of approximating a continuous distribution with a simple discrete distribution?
The following problem came up on the Mathematica forum as "Generating a list of integers that roughly satisfy a distribution": Given $n$, find $n$ integers (possibly with duplicates) whose ...
2
votes
0answers
42 views
Is this exponential family Gaussian?
Let $m$ be a positive measure on the real line. Assume that $\exp k(t)=\int\exp(xt)m(dx)<\infty$ for $a<t<b$ and that $\exp k(t)=\infty$ if $t\notin[a,b],$, with $-\infty\leq a<b\leq ...
2
votes
1answer
105 views
Computation complexity of calculating the cdf of an n-th dimensional gaussian random vector
Suppose you have a general $n$-th dimensional random Gaussian vector with probability distribution function $\mathcal{N}\left(\mathbf{x}|\boldsymbol{\mu},\boldsymbol{\Sigma}\right)$.
What is the ...
2
votes
1answer
80 views
Maximal component of a multivariate Gaussian distribution
Suppose you have a general random Gaussian vector $\mathbf{X}\sim\mathcal{N}\left(\boldsymbol{\mu},\boldsymbol{\Sigma}\right)$. I'm looking for the simple way to calculate the distribution of the ...
0
votes
0answers
67 views
Obtaining the 'threshold' of a distribution
Context of Research
Consider the expression:
\begin{align}
\widehat{\Theta}(\rho)_i = \frac{1}{(1-\rho)\Delta t} \ln\left(\frac{1}{T} \sum_{t=1}^T \left(\frac{1+r_t}{1+rf_t}\right)^{1-\rho} \right) ...
0
votes
0answers
56 views
Eigen value distribution of autocorrelated Wishart matrix
Suppose the matrix W is constructed as $W=XX^T$ where $X_i(t) = \phi_i X_i(t-1) + a_i(t)$, and $a_i(t)$ ~ $N(0,1)$. I am interested in knowing the eigen value distribution of W. My google search on ...
3
votes
3answers
246 views
Repeated draws from multinomial distribution
(This is a cross-post from Math StackExchange http://math.stackexchange.com/questions/609641/multinomial-distribution-sum-of-squared-probabilities)
Let $\vec X = (X_1, \dots, X_k)$ be a draw from a ...
0
votes
0answers
45 views
Relative influences of terms in the law of total cumulance
$\newcommand{\cum}{\operatorname{cum}}$
Denote the joint cumulant of several random variables by $\cum(A,B,C,\ldots)$ (more precisely, the cumulant of the joint probability distribution). In ...
2
votes
2answers
264 views
What is the maximum entropy distribution on the natural numbers?
On the reals $\mathbb{R}$, the maximum entropy distribution with a given mean and variance is the Gaussian distribution.
Let $\mu, \sigma > 0$. What is the maximum entropy distribution on the ...
2
votes
0answers
35 views
existence of Markov operators not generated by transition probability function
Transition probability functions can always be used to generate Markov operators, correct? So is it correct to say that a Markov process is a collection of Markov operators? On the other hand, are ...
0
votes
0answers
40 views
Distinguishing two different matrix distributions in polynomial time
I have two distributions:
$\{ (f^TA + e_1, f^T(As+e) \}$ and $\{ (f^TA, f^T(As+e) + s_i \}$
where $A$ is a randomly generated $m \times n$ binary matrix $A, A_{ij} \in \{0,1\}$, $f$ and $e$ are a ...
0
votes
0answers
41 views
Estimating the mean of a bivariate distribution by randomly sampling variates and rounding them to integer coordinates
Imagine that we have a particle sampling positions on a two-dimensional plane according to a bivariate probability distribution: $A*e^{-(\frac{(x-x_0)}{2\sigma_x^2}+\frac{(y-y_0)}{2\sigma_y^2})}$, ...
1
vote
0answers
105 views
Sampling a two-dimensional Gaussian distribution at points along an integer lattice
Please consider a two-dimensional Gaussian of the general form: $A*e^{-(\frac{(x-x_0)}{2\sigma_x^2}+\frac{(y-y_0)}{2\sigma_y^2})}$, where $C$ is the peak of the Gaussian, i.e. the point at which the ...
0
votes
1answer
243 views
Distribution of the sum of independent normal and gamma random variables
Let $Z = X + Y$ where $X \sim N\left(\mu, \sigma^2 \right)$ and $Y \sim \Gamma\left(k, \theta \right)$ using this parametrization of the Gamma distribution. Also assume $X$ and $Y$ are independent. ...
2
votes
2answers
157 views
What's the probability of differences among n independent uniform distribution variables?
Given n independent random variables $x_1,x_2,...,x_n$, they have standard uniform distributions over [0,1]. Then what's the probability that there is at least one $|x_i-x_j| >= d$ for any ...
2
votes
2answers
112 views
Unusual Differential Equation for CDF
Consider the following differential equation
$$F(cx) = F(x) + x F'(x)$$
for $c>1$.
Does this differential equation belong to a some well known class?
Is there a way to find all the solutions ...
1
vote
0answers
62 views
Determining the position of a coordinate by binning Gaussian noise around that coordinate to lattice points with vertex-specific probabilities [closed]
(NOTE: I have changed and hopefully simplified this question by removing the section on randomly perturbing lattice points, and instead specifying that the counts at each vertex should be randomly ...
1
vote
0answers
56 views
Multiple independent random number streams
This question is somehow related to this one.
Having multiple streams of pseudo-random numbers known to be independent and with a uniform distribution (U1, U2,...,Un) I want to do Monte Carlo ...
0
votes
1answer
110 views
Distribution and moments of ratio of two beta variables?
If $X$ and $Y$ are two Beta random variables, I am interested in the distribution of their ratio $X/Y$. More specifically, I am interested in the moment generating function of this ratio. There is a ...
2
votes
0answers
66 views
Quantifying the “flatness” of functions which are the Fourier transforms of positive functions
Short version of question: I'm trying to understand the extent to which a function is prevented from being "flat" as a result of being the Fourier transform of a positive function. That is, the extent ...
0
votes
0answers
51 views
Distribution of Quadratic Forms in Rademacher Random Vector
This is a fairly well-known theorem on quadratic form in normal distribution, which is taken from here.
If $\mathbf y ∼ N(0, \sigma^2)$ is an $n\times 1$ column vector and $\mathbf M$ is an ...
1
vote
1answer
135 views
Set of distributions that minimize KL divergence,
Assuming that $p,q$ are probability distributions defined on the same support $\{x_i\}_{0 \leq i \leq n}$, $\epsilon$ a small real number, and $D_{KL}$ the Kullback-Leibler divergence,
is there a ...
