Benktander type II distribution

Probability density function
Cumulative distribution function
Parameters	${\displaystyle a>0}$ (real) ${\displaystyle 0<b\leq 1}$ (real)
Support	${\displaystyle x\geq 1}$
PDF	${\displaystyle e^((\frac {a}{b))(1-x^{b})}x^{b-2}\left(ax^{b}-b+1\right)}$
CDF	${\displaystyle 1-x^{b-1}e^((\frac {a}{b))(1-x^{b})))$
Mean	${\displaystyle 1+{\frac {1}{a))}$
Median	${\displaystyle {\begin{cases}{\frac {\log(2)}{a))+1&{\text{if))\ b=1\\\left(\left({\frac {1-b}{a))\right)\mathbf {W} \left({\frac {2^{\frac {b}{1-b))ae^{\frac {a}{1-b))}{1-b))\right)\right)^{\tfrac {1}{b))&{\text{otherwise))\ \end{cases))}$ Where ${\displaystyle \mathbf {W} (x)}$ is the Lambert W function[note 1]
Mode	${\displaystyle 1}$
Variance	${\displaystyle {\frac {-b+2ae^{\frac {a}{b))\mathbf {E} _{1-{\frac {1}{b))}\left({\frac {a}{b))\right)}{a^{2}b))}$ Where ${\displaystyle \mathbf {E} _{n}(x)}$ is the generalized Exponential integral[note 1]

The Benktander type II distribution, also called the Benktander distribution of the second kind, is one of two distributions introduced by Gunnar Benktander (1970) to model heavy-tailed losses commonly found in non-life/casualty actuarial science, using various forms of mean excess functions (Benktander & Segerdahl 1960). This distribution is "close" to the Weibull distribution (Kleiber & Kotz 2003).