# Tagged Questions

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 ...
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 ...
131 views

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, ...
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. ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
197 views

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 ...
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 ...