Benktander-Weibull heavy-tailed distribution on x >= 1
BenktanderWeibullDistribution[a, b]
Mean[BenktanderWeibullDistribution[a, b]]
→ 1 + a^(-1)PDF[BenktanderWeibullDistribution[a, b], x]
→ Piecewise[{{E^((a*(1 - x^b))/b)*x^(-2 + b)*(1 - b + a*x^b), x >= 1}}, 0]CDF[BenktanderWeibullDistribution[a, b], x]
→ Piecewise[{{1 - E^((a*(1 - x^b))/b)*x^(-1 + b), x >= 1}}, 0]