Benktander Gibrat distribution

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Benktander distribution of the first kind
Parameters a>0 (real)
b>0 (real
Support x>1
pdf  e^{-b {\mathrm{Log}[x]}^2}x^{-2-a} \left( -\tfrac{2b}{a} + \left(1 + a + 2b \mathrm{Log}[x]\right)\left(1+\tfrac{2b \mathrm{Log}[x]}{a}\right)\right)
CDF  1 -  e^{-b {\mathrm{Log}[x]}^2} x^{-1-a} \left(1+\tfrac{2b \mathrm{Log}[x]}{a}\right)
Mean 1+\tfrac{1}{a}
Variance  \frac{-\sqrt{b}+a e^{\tfrac{(-1+a)^2}{4b}} \sqrt{\pi} \operatorname{erfc}\left(\tfrac{-1+a}{2\sqrt{b}}\right) }{a^2\sqrt{b}}

Benktander distribution of the first kind

Related distributions[edit]

 \lim_{b \to 0} \mathrm{Benktander}(a,b) \sim \mathrm{Pareto}(1,a+1) .


  • Kleiber, Christian; Kotz, Samuel (2003) Statistical size distributions in economics and actuarial sciences, Wiley-Interscience, ISBN 0-471-15064-9
  • Benktander, G.; Seherdahl, C.O. (1960) "On the analytical representation of claim distributions with special reference to excess-of-loss reinsurance". Trans. 16-th Intern. Congress Actuaries, 626-646.