## 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 $\dfrac{\alpha-1}{\beta+1}$