Tuesday, 14 May 2013

Distributions processed by SciStatCalc

Below is a table in which are listed the 17 distributions whose CDF and inverse-CDF (also known as the quantile function) are computed by SciStatCalc. The table is embedded as a PDF file - feel free to download it. The table could come in handy as a quick reference guide for some attributes of the most popular distributions. The expression of the CDF occurs in the right most column.

 

Examining the table above, $I_x(a,b)$ denotes the normalised incomplete Beta function:-
$I_{x}(a,b)=\frac{B_x(a,b)}{B(a,b)}$
where
$B_x(a,b)=\int_0^xt^{a-1}(1-t)^{b-1}dt$
and
$B(a,b)=\int_0^1t^{a-1}(1-t)^{b-1}dt$

The lower incomplete Gamma function $\gamma(a,x)$ is evaluated as:-
$\gamma(a,x)=\int_0^xt^{a-1}e^{-t}dt$
The Gamma function $\Gamma(a)$ is:-
$\Gamma(a)=\int_0^{\infty}t^{a-1}e^{-t}dt$

The error function $erf(x)$ is:-
$erf(x)=\frac{2}{\sqrt{\pi}}\int_0^xexp{(-t^2)}dt$