Beta Distribution

$\begin{figure}\begin{center}\BoxedEPSF{BetaDistribution.epsf scaled 650}\end{center}\end{figure}$

A general type of statistical Distribution which is related to the Gamma Distribution. Beta distributions have two free parameters, which are labeled according to one of two notational conventions. The usual definition calls these $\alpha$ and $\beta$ , and the other uses $\beta'\equiv\beta-1$ and $\alpha'\equiv\alpha-1$ (Beyer 1987, p. 534). The above plots are for $(\alpha, \beta)=(1,1)$ [solid], (1, 2) [dotted], and (2, 3) [dashed]. The probability function and Distribution Function are given by

$\displaystyle P(x)$	$\textstyle =$	$\displaystyle {(1-x)^{\beta-1}x^{\alpha-1}\over B(\alpha,\beta)}$
	$\textstyle =$	$\displaystyle {\Gamma(\alpha +\beta)\over \Gamma(\alpha)\Gamma(\beta)} (1-x)^{\beta-1} x^{\alpha-1}$	(1)
$\displaystyle D(x)$	$\textstyle =$	$\displaystyle I(x; a, b),$	(2)

where

is the Beta Function,

is the Regularized Beta Function, and

where $\alpha$ , $\beta>0$ . The distribution is normalized since

$\displaystyle \int_0^1 P(x)\,dx$	$\textstyle =$	$\displaystyle {\Gamma(\alpha +\beta)\over \Gamma(\alpha)\Gamma(\beta)} \int_0^1 x^{\alpha-1} (1-x)^{\beta-1}\,dx$	(3)
	$\textstyle =$	$\displaystyle {\Gamma(\alpha+\beta)\over\Gamma(\alpha)\Gamma(\beta)} B(\alpha,\beta)= 1.$	(4)

The Characteristic Function is

$\begin{displaymath} \phi(t)={}_1F_1(a,a+b,it). \end{displaymath}$

(5)

The Moments are given by

$\begin{displaymath} M_r=\int_0^1 P(x)(x-\mu)^r\,dx = {\Gamma(\alpha+\beta)\Gamma(\alpha+r)\over\Gamma(\alpha+\beta+r)\Gamma(\alpha)}. \end{displaymath}$

(6)

The Mean is

$\displaystyle \mu$	$\textstyle =$	$\displaystyle {\Gamma(\alpha +\beta)\over \Gamma(\alpha)\Gamma(\beta)}\int_0^1 x^{\alpha-1}(1-x)^{\beta-1} x\,dx$
	$\textstyle =$	$\displaystyle {\Gamma(\alpha +\beta)\over \Gamma(\alpha)\Gamma(\beta)} B(\alpha+1,\beta)$
	$\textstyle =$	$\displaystyle {\Gamma(\alpha +\beta)\over \Gamma(\alpha)\Gamma(\beta)} {\Gamma(... ... +1)\Gamma(\beta)\over \Gamma(\alpha +\beta +1)} = {\alpha\over \alpha +\beta},$	(7)

and the Variance, Skewness, and Kurtosis are

$\displaystyle \sigma^2$	$\textstyle =$	$\displaystyle {\alpha\beta\over (\alpha +\beta)^2(\alpha +\beta +1)}$	(8)
$\displaystyle \gamma_1$	$\textstyle =$	$\displaystyle {2(\sqrt{\beta}-\sqrt{\alpha}\,)(\sqrt{\alpha}+\sqrt{\beta}\,)\sqrt{1+\alpha+\beta}\over \sqrt{\alpha\beta}\,(\alpha+\beta+2)}$	(9)
$\displaystyle \gamma_2$	$\textstyle =$	$\displaystyle {6(\alpha^2+\alpha^3-4\alpha\beta-2\alpha^2\beta+\beta^2-2\alpha\beta^2+\beta^3)\over \alpha\beta(\alpha+\beta+2)(\alpha+\beta+3)}.$
			(10)

The Mode of a variate distributed as $\beta(\alpha,\beta)$ is

$\begin{displaymath} \hat x = {\alpha-1\over \alpha+\beta-2}. \end{displaymath}$

(11)

In ``normal'' form, the distribution is written

$\begin{displaymath} f(x)={\Gamma(\alpha+\beta)\over\Gamma(\alpha)\Gamma(\beta)} x^{\alpha-1}(1-x)^{\beta-1} \end{displaymath}$

(12)

and the Mean, Variance, Skewness, and Kurtosis are

$\displaystyle \mu$	$\textstyle =$	$\displaystyle {\alpha\over\alpha+\beta}$	(13)
$\displaystyle \sigma^2$	$\textstyle =$	$\displaystyle {\alpha\beta\over(\alpha+\beta)^2(1+\alpha+\beta)}$	(14)
$\displaystyle \gamma_1$	$\textstyle =$	$\displaystyle {2(\sqrt{\alpha}-\sqrt{\beta}\,)(\sqrt{\alpha}+\sqrt{\beta}\,)\sqrt{1+\alpha+\beta}\over \sqrt{\alpha\beta} (\alpha+\beta+2)}$	(15)
$\displaystyle \gamma_2$	$\textstyle =$	$\displaystyle {3(1+\alpha+\beta)(2\alpha^2-2\alpha\beta+\alpha^2\beta+2\beta^2+\alpha\beta^2)\over \alpha\beta(\alpha+\beta+2)(\alpha+\beta+3)}.$
			(16)

See also Gamma Distribution

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 944-945, 1972.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 534-535, 1987.