Log-gamma distribution with shape scale and location parameters
LogGammaDistribution[α, β, μ]
Mean[LogGammaDistribution[α, β, μ]]
→ Piecewise[{{-1 + (1 - β)^(-α) + μ, β < 1}}, Infinity]PDF[LogGammaDistribution[α, β, μ], x]
→ Piecewise[{{Log[1 + x - μ]^(-1 + α)/(β^α*(1 + x - μ)^((1 + β)/β)*Gamma[α]), x >= μ}}, 0]CDF[LogGammaDistribution[α, β, μ], x]
→ Piecewise[{{GammaRegularized[α, 0, Log[1 + x - μ]/β], x >= μ}}, 0]