Pearson Type VI

The density function of the Pearson Type VI distribution is

\[  f(x) = \frac{\left(\frac{x}{\beta }\right)^{\alpha _1 - 1}}{\beta G(\alpha _1, \alpha _2)\left[1 + \left(\frac{x}{\beta }\right)\right]^{\alpha _1 + \alpha _2}}  \]

where $x > 0$ and

\[  G(\alpha _1, \alpha _2) = \frac{\Gamma (\alpha _1)\Gamma (\alpha _2)}{\Gamma (\alpha _1 + \alpha _2)}  \]

The function $\Gamma (z)$ is defined in the section Beta.

Parameters:

$\alpha _1$

is a shape parameter, $\alpha _1 > 0$.

$\alpha _2$

is a shape parameter, $\alpha _2 > 0$.

$\beta $

is a scale parameter, $\beta > 0$.

If $X_1$ and $X_2$ are independent random variables with $X_1 \sim \mbox{Gamma}(\alpha _1, \beta )$ and $X_2 \sim \mbox{Gamma}(\alpha _2, 1)$, then $Y = \frac{X_1}{X_2} \sim \mbox{PearsonTypeVI}(\alpha _1, \alpha _2, \beta )$.