0
votes
0answers
129 views

Is there a way to find a steady state probability density for a given transformation?

I was looking at probability density transformations here. They use $g(y) = f(x(y))|dx/dy|$ where $x(y)$ is the inverse of the transformation. Is there a way to, given only the transformation, find ...
0
votes
0answers
20 views

Moments of Multivariate Distributions

I would like to be able to compute higher moments for multivariate distributions. I have seen the results presented by Isserlis's theorem which shows how to do this for the multivariate normal, but ...
2
votes
1answer
131 views

Sum of two gamma/Erlang random variables $\Gamma(m,\lambda)$ and $\Gamma(n, \mu)$ with integer numbers $m \neq n, \lambda \neq \mu$

The gamma distribution with parameters $m > 0$ and $\lambda > 0$ (denoted $\Gamma(m, \lambda)$) has density function $$f(x) = \frac{\lambda e^{-\lambda x} (\lambda x)^{m - 1}}{\Gamma(m)}, x > ...
2
votes
0answers
45 views

A Lemma in the book “ Mathematical Method for financial markets” (Chapter 5, Section 5.7)

In page 307, Section 5.7, Chapter 5 of the book "mathematical methods for financial markets" by Jeanblanc, Yor and Chesney, Lemma 5.7.1 is given as follows: Lemma 5.7.1.1 Let $W$ be a Brownian ...
0
votes
0answers
28 views

Learning resources for Probability Distributions/Models

I've a good background in basic probability. I need to learn and get a good grip on the probability distributions and stochastic processes, counting processes, and other related topics. I am already ...
0
votes
2answers
45 views

Probability distribution in wireless channel?

Let suppose that I have a random variable $X_{mn}=\sqrt{\left(1/d_{mn}\right)^\alpha}\times h_{mn}$ wherr $d_{mn}$ is a random variable with uniform distribution and $h_{mn}$ is a random variable with ...
3
votes
1answer
52 views

Name/significance of integral of the square of a probability density function

Background/Motivation Given a probability density function $f(x)$, the mean of the corresponding random variable is the $x$-coordinate of the centroid of the region under the graph of $f$. I ...
0
votes
0answers
27 views

$(X, Y)^T$ ~ multivariate normal does NOT implies that X | Y $\in$ [a,b] has normal distribution??

I just found a rather surprising fact about the multivariate normal distribution. Suppose $(X,Y)^T$ has bivariate normal distribution; $$ \begin{bmatrix} X\\ Y \end{bmatrix} \sim MVN_2 \Big( ...
0
votes
0answers
13 views

Change of variables reference

I am looking for a good (comprehensive) reference covering the computation of PDFs under change of variables. I have found some formulas in Wikipedia, for example univariate and multivariate formulas, ...
0
votes
0answers
20 views

Systematic elementary exposition of multivariate c.d.f.s?

An example of some initially counterintuitive things that can happen with the joint cumulative probability distribution function of a tuple of real-valued random variables is found in this question. ...
1
vote
2answers
84 views

What is this mathematics sub-field called?

I would love to answer another question on this site, but I am totally unfamiliar with the required technique. I mean, I don't even know the sub-field's name. The field I am looking for is one that ...
2
votes
0answers
66 views

Maximum bin load for $\alpha n$ balls into $n$ bins

In a paper I am reading the author writes: A standard result concerning balls and bins shows that if we throw at least $\alpha n$ balls into at most $n$ bins, then the maximum bin load is ...
2
votes
2answers
46 views

A Reference Book Justifying Different Distributions

Well I am trying to find a book that could come up with a rationale behind different distributions but not only defining them and giving an intuition about the structure of distributions. For example ...
0
votes
0answers
118 views

fractional moments of binomial distribution

I would appreciate your help in learning about the quantity $$ \large\sum_{j=0}^J {J \choose j} p^j (1-p)^{J-j} j^\alpha, $$ for any $\alpha > 0$, but in particular for $\alpha\in (0,1)$. Which ...
0
votes
0answers
79 views

Generalized Rayleigh distribution?

A variable $z=\sqrt{x^2+y^2}$ is Rayleigh distributed with parameter $\sigma$ if $x\sim\mathcal{N}(0,\sigma^2)$ and $y\sim\mathcal{N}(0,\sigma^2)$. Is there a known/named distribution for the case ...
1
vote
0answers
42 views

Bernstein type inequalities. Is there a standard list?

Suppose I have a sequence of iid random variables $X_i\geq 0$ with mean $\mu$ and $\mathbb E \left(e^{tX_i}\right) = G(t)$. Set $$S_n = \sum_{i=1} X_n.$$ For the purpose of this question the ...
2
votes
0answers
37 views

Inferring a probability distribution from another probability distribution

Let $A$ and $B$ be real-valued random variables, with $f_A$ and $f_B$ their probability density functions. Let's say we can observe the values of $A$ many times and estimate $f_A$ fairly precisely. We ...
1
vote
0answers
62 views

What is known about $n$ independent random variables that yield a “converse” to uniform sample of a coordinate from a surface of an $n$-sphere?

It's well-known that to sample a coordinate $(Y_1,\ldots,Y_n)$ from a surface of an $n$-dimensional unit-radius sphere, it suffices to generate $n$ independent random variables $X_1,\ldots,X_n$ from ...
3
votes
0answers
197 views

What is a spherical Gaussian kernel?

In this paper (page 8, Example 3), a spherical Gaussian kernel is defined by the formula $$K(\mathrm x,\mathrm y)=e^{-2\epsilon(1- \mathrm x\cdot\mathrm y)}$$ where $\mathrm{x,y}\in ...
1
vote
0answers
37 views

Large $k$ asymptotic of $\Pr(X=k)$ for a compound Poisson random variable $X$

Let $N \sim \operatorname{Poisson}(\mu)$, and let $X|N = \sum_{k=1}^N Y_k$, where $Y_k$ are iid non-negative integer-valued random variables. The distribution of $X$ is known as compound Poisson ...
1
vote
2answers
101 views

Reference on Polynomial Chaos

I need to understand the basics of "Polynomial Chaos" (http://en.wikipedia.org/wiki/Polynomial_chaos), and I'm having trouble finding a good reference on it. I'm looking for something rigorous enough ...
1
vote
1answer
59 views

Reference about Fredholm determinants

I am searching for a reference book on Fredholm determinants. I am mainly interested in applications to probability theory, where cumulative distribution functions of limit laws are expressed in terms ...
1
vote
1answer
82 views

Sampling and probability generating functions - reference wanted

Suppose I have a huge (effectively infinite) population of widgets. The number of widgets that are broken is given by a random variable $X$, whose probability generating function is $p(z) = E(z^X)$. ...
1
vote
1answer
182 views

Well known probability distributions defined on a $n$-dimensional simplex besides the Dirichlet distribution?

Are there well known probability distributions defined on a n-dimensional simplex besides the Dirichlet distribution where the variation of of each component doesn't vary as much when the mean of the ...
0
votes
1answer
51 views

Is there a name for this family of probability distributions?

I am wondering whether a family of probability distributions with the following form of a density function has a name: $$f(x)=C*\operatorname{Exp}(-B|x|^A)$$ where $A$, $B$ and $C$ are positive ...
-2
votes
2answers
117 views

Covariance 's relationship with pure math and probabilty? [closed]

I've been looking up a lot of statistical books and cannot find out mathmatical insight behind it, but my math level wasn't allow me to read the mathmatical statistics books and get the math behind ...
3
votes
1answer
75 views

Is this a known distribution?

I came across the distribution on $(0,1]$ with the following density function $$f(x) = \frac{2}{\pi}\sqrt{\frac{1}{x}-1}$$ Is this a known distribution? Any references will be appreciated.
2
votes
1answer
557 views

Joint distribution of two functions of two random variables

1) Suppose I have two random variables $A > 0$ and $B > 0$ with joint p.d.f. $f_{A,B}(a,b)$, and two random variables $X = g_1(A,B)$ and $Y = g_2(A,B)$. What is the general procedure for ...
2
votes
2answers
662 views

Computation of the probability density function for $(X,Y) = \sqrt{2 R} ( \cos(\theta), \sin(\theta))$

Let $R$ be a almost surely non-negative continuous random variable with absolutely continuous measure, and $\Theta$ be an independent random variable, uniformly distributed on the interval $[0, 2 ...
3
votes
2answers
185 views

Determining distribution of $X_t = \int_0^t W_s^2 \mathrm{d} s$

Premise Let $W_t$ be the standard Wiener process, and let $X_t = \int_0^t W_s^2 \mathrm{d} s$. I am interested in determining the distribution of $X_t$. What I did My line of attack has been to ...
2
votes
1answer
126 views

cumulants and infinite divisibility

Where might I find a clear exposition of how to prove that a real-valued probability distribution for which all moments exist is infinitely divisible if and only if all of its cumulants of even order ...
2
votes
3answers
170 views

Are magic squares inevitable?

Consider: If we are given a reasonably well-behaved statistical population of real numbers, then samples that are not small will have mean approximately equal to that of the population, right? Then: ...
1
vote
1answer
133 views

good reference to kernel density estimation

Please suggest a good reference for introduction to Kernel Density Estimation. I am an electrical engineer and i want some basic introduction so that i can get to start using it and also understand ...