Generalized normal (exponential power) distribution
ExponentialPowerDistribution[κ, μ, σ]
ExponentialPowerDistribution[κ]
Mean[ExponentialPowerDistribution[κ, μ, σ]]
→ μPDF[ExponentialPowerDistribution[κ, μ, σ], x]
→ Piecewise[{{1/(2*E^(((x - μ)/σ)^κ/κ)*κ^κ^(-1)*σ*Gamma[1 + κ^(-1)]), x >= μ}}, 1/(2*E^(((-x + μ)/σ)^κ/κ)*κ^κ^(-1)*σ*Gamma[1 + κ^(-1)])]CDF[ExponentialPowerDistribution[κ, μ, σ], x]
→ Piecewise[{{GammaRegularized[κ^(-1), ((-x + μ)/σ)^κ/κ]/2, x < μ}}, 1 - GammaRegularized[κ^(-1), ((x - μ)/σ)^κ/κ]/2]