1
vote
1answer
31 views

Central limits without replacement in a finite population.

"Everybody knows" that there are lots of variations on the theme of the central limit theorem. The most frequently seen form seems to be this: Suppose $X_1,X_2,X_3,\ldots$ are i.i.d. random variables ...
0
votes
0answers
135 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
26 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
185 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
51 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
33 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
46 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
65 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
30 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
14 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
21 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
89 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
72 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
48 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 ...
1
vote
0answers
48 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
38 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
63 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
239 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
38 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
117 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
65 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
88 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
201 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
52 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 ...
3
votes
1answer
76 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
602 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
747 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
191 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
130 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
175 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
141 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 ...