Johnson system of distributions with four families (SB/SL/SN/SU)
JohnsonDistribution["SB", γ, δ, μ, σ]
JohnsonDistribution["SL", γ, δ, μ, σ]
JohnsonDistribution["SU", γ, δ, μ, σ]
JohnsonDistribution["SN", γ, δ, μ, σ]
Mean[JohnsonDistribution["SL", γ, δ, μ, σ]]
→ μ + E^((1 - 2*γ*δ)/(2*δ^2))*σMean[JohnsonDistribution["SN", γ, δ, μ, σ]]
→ (δ*μ - γ*σ)/δPDF[JohnsonDistribution["SB", γ, δ, μ, σ], x]
→ Piecewise[{{(δ*σ)/(E^((γ + δ*Log[(-μ + x)/(μ + σ - x)])^2/2)*Sqrt[2*Pi]*(μ + σ - x)*(-μ + x)), μ < x < μ + σ}}, 0]