Exponential gamma distribution (log of a gamma variate) with shape scale and location
ExpGammaDistribution[κ, θ, μ]
Mean[ExpGammaDistribution[κ, θ, μ]]
→ μ + θ*PolyGamma[0, κ]CDF[ExpGammaDistribution[κ, θ, μ], x]
→ GammaRegularized[κ, 0, E^((x - μ)/θ)]Variance[ExpGammaDistribution[κ, θ, μ]]
→ θ^2*PolyGamma[1, κ]