Benktander distribution of the first kind

Parameters	${\displaystyle a>0}$ (real) ${\displaystyle 0<b\leq {\frac {a(a+1)}{2))}$ (real)
Support	${\displaystyle x\geq 1}$
PDF	${\displaystyle \left(\left[\left(1+{\frac {2b\log x}{a))\right)\left(1+a+2b\log x\right)\right]-{\frac {2b}{a))\right)x^{-\left(2+a+b\log x\right)))$
CDF	${\displaystyle 1-\left(1+{\frac {2b}{a))\log x\right)x^{-\left(a+1+b\log x\right)))$
Mean	${\displaystyle 1+{\tfrac {1}{a))}$
Variance	${\displaystyle {\frac {-{\sqrt {b))+ae^{\frac {(a-1)^{2)){4b)){\sqrt {\pi ))\;{\textrm {erfc))\left({\frac {a-1}{2{\sqrt {b))))\right)}{a^{2}{\sqrt {b))))}$[note 1]

The Benktander type I distribution 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). The distribution of the first type is "close" to the log-normal distribution (Kleiber & Kotz 2003).