Benktander-Gibrat actuarial distribution
BenktanderGibratDistribution[a, b]
Mean[BenktanderGibratDistribution[a, b]]
→ 1 + a^(-1)PDF[BenktanderGibratDistribution[a, b], x]
→ Piecewise[{{(x^(-2 - a)*((-2*b)/a + (1 + a + 2*b*Log[x])*(1 + (2*b*Log[x])/a)))/E^(b*Log[x]^2), x >= 1}}, 0]CDF[BenktanderGibratDistribution[a, b], x]
→ Piecewise[{{1 - (x^(-1 - a)*(1 + (2*b*Log[x])/a))/E^(b*Log[x]^2), x >= 1}}, 0]