Pearson Type III Distribution

A skewed distribution which is similar to the Binomial Distribution when $p\not=q$ (Abramowitz and Stegun 1972, p. 930).

$\begin{displaymath} y=k(t+A)^{A^2-1}e^{-At}, \end{displaymath}$

(1)

for $t\in[0,\infty)$ where

$\displaystyle A$	$\textstyle \equiv$	$\displaystyle 2/\gamma$	(2)
$\displaystyle K$	$\textstyle \equiv$	$\displaystyle {A^{A^2}e^{-A^2}\over \Gamma(A^2)},$	(3)

$\Gamma(z)$ is the Gamma Function, and

is a standardized variate. Another form is

$\begin{displaymath} P(x)={1\over\beta\Gamma(p)} \left({x-\alpha\over\beta}\right... ...1}\mathop{\rm exp}\nolimits \left({x-\alpha\over\beta}\right). \end{displaymath}$

(4)

For this distribution, the Characteristic Function is

$\begin{displaymath} \phi(t)=e^{i\alpha t}(1-i\beta t)^{-p}, \end{displaymath}$

(5)

and the Mean, Variance, Skewness, and Kurtosis are

$\displaystyle \mu$	$\textstyle =$	$\displaystyle \alpha+p\beta$	(6)
$\displaystyle \sigma^2$	$\textstyle =$	$\displaystyle p\beta^2$	(7)
$\displaystyle \gamma_1$	$\textstyle =$	$\displaystyle {2\over\sqrt{p}}$	(8)
$\displaystyle \gamma_2$	$\textstyle =$	$\displaystyle {6\over p}.$	(9)

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.