Beta Type II Distribution
A continuous random variable $X$ is said to have a beta type II distribution with parameters $\alpha$ and $\beta$ if its p.d.f. is given by
$$ \begin{equation*} f(x)=\left\{ \begin{array}{ll} \frac{1}{B(\alpha,\beta)}\cdot\frac{x^{\alpha-1}}{(1+x)^{\alpha+\beta}}, & \hbox{$0\leq x\leq\infty$;} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{equation*} $$
where,
-
$B(\alpha,\beta) =\frac{\Gamma \alpha \Gamma \beta}{\Gamma (\alpha+\beta)}=\int_0^1 x^{\alpha-1}(1-x)^{\beta-1}\; dx$is a beta function and -
$\Gamma \alpha$is a gamma function.
Mean of Beta Type II Distribution
The mean of beta type II distribution is $E(X) = \dfrac{\alpha}{\beta-1}$.
Variance of Beta Type II Distribution
The variance of beta type II distribution is $V(X) = \dfrac{\alpha(\alpha+\beta-1)}{(\beta-1)^2(\beta-2)}$.
Harmonic Mean of Beta Type II Distribution
The harmonic mean of beta type II distribution is $H = \dfrac{\alpha-1}{\beta}$.
Mode of Beta Type-II Distribution
The mode of beta type II distribution is $\dfrac{\alpha-1}{\beta+1}$
