Benini log-quadratic hazard distribution
BeniniDistribution[α, β, σ]
Mean[BeniniDistribution[α, β, σ]]
→ σ + (E^((-1 + α)^2/(4*β))*Sqrt[Pi]*σ*Erfc[(-1 + α)/(2*Sqrt[β])])/(2*Sqrt[β])PDF[BeniniDistribution[α, β, σ], x]
→ Piecewise[{{(α/x + (2*β*Log[x/σ])/x)/(E^(β*Log[x/σ]^2)*(x/σ)^α), x >= σ}}, 0]CDF[BeniniDistribution[α, β, σ], x]
→ Piecewise[{{1 - 1/(E^(β*Log[x/σ]^2)*(x/σ)^α), x >= σ}}, 0]