P.Mean: Statistical distributions borrowed from Wikipedia (created 2012-05-02). News: Sign up for "The Monthly Mean," the newsletter that dares to call itself average, www.pmean.com/news.

I am finding that I need to refer often to various statistical distributions on Wikipedia. Since Wikipedia is open source, I can re-use their content here. So here are some of my favorite statistical distributions.

Some composite distributions involving the beta distribution are

Beta negative binomial distribution.

Beta binomial distribution.

A closely related distribution is Inverse beta distribution.

There is an application where I needed the moment generating function of the beta distribution. Here it is.

$1 +\sum_{k=1}^{\infty} \left( \prod_{r=0}^{k-1} \frac{\alpha+r}{\alpha+\beta+r} \right) \frac{t^k}{k!}$

$\frac{1}{2^{\frac{k}{2}}\Gamma\left(\frac{k}{2}\right)}\; x^{\frac{k}{2}-1} e^{-\frac{x}{2}}\,$

A closely related distribution is the noncentral chi-square distribution.

$\frac{1}{2}e^{-(x+\lambda)/2}\left (\frac{x}{\lambda} \right)^{k/4-1/2} I_{k/2-1}(\sqrt{\lambda x})$

This distribution has two forms.

F distribution.

$\frac{\sqrt{\frac{(d_1\,x)^{d_1}\,\,d_2^{d_2}} {(d_1\,x+d_2)^{d_1+d_2}}}} {x\,\mathrm{B}\!\left(\frac{d_1}{2},\frac{d_2}{2}\right)}\!$

A closely related distribuiton is the noncentral F distribution.

$p(f) =\sum\limits_{k=0}^\infty\frac{e^{-\lambda/2}(\lambda/2)^k}{ B\left(\frac{\nu_2}{2},\frac{\nu_1}{2}+k\right) k!} \left(\frac{\nu_1}{\nu_2}\right)^{\frac{\nu_1}{2}+k} \left(\frac{\nu_2}{\nu_2+\nu_1f}\right)^{\frac{\nu_1+\nu_2}{2}+k}f^{\nu_1/2-1+k}$

This distribution has two forms.

A closely related distribution is the Inverse Gamma distribution.

$\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)$

Geometric distribution.

Hypergeometric distribution.

Negative binomial distribution.

t distribution.

Other resources

Faming Liang, a Professor of Statistics at Texas A&M Univeristy, has some nicely written course notes in PDF format on various distributions as part of his handouts for the STAT 610 course. This includes a derivation of the link between the cumulative probability of a Poisson random variable and percentiles from the Gamma distribution.