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Pearson Type III Distribution

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

\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 $t$ is a standardized variate. Another form is
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
\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)


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

© 1996-9 Eric W. Weisstein