Tukey lambda quantile-defined distribution
TukeyLambdaDistribution[λ]
TukeyLambdaDistribution[λ, μ, σ]
TukeyLambdaDistribution[{λ 1, λ 2}, μ, {σ 1, σ 2}]
Mean[TukeyLambdaDistribution[λ]]
→ Piecewise[{{0, λ > -1}}, Indeterminate]Variance[TukeyLambdaDistribution[λ]]
→ Piecewise[{{(2*(2*λ)! - 2*λ!^2)/(λ^2*(1 + 2*λ)!), λ > -1/2}}, Indeterminate]Plot[Table[PDF[TukeyLambdaDistribution[λ], x], {λ, {-1, 0.5, 1}}]//Evaluate, {x, -3, 3}, Filling -> Axis]
→ -Graphics-