Hyperbolic distribution with parameters for shape and location
HyperbolicDistribution[α, β, δ, μ]
HyperbolicDistribution[λ, α, β, δ, μ]
Mean[HyperbolicDistribution[α, β, δ, μ]]
→ μ + (β*δ*BesselK[2, Sqrt[α^2 - β^2]*δ])/(Sqrt[α^2 - β^2]*BesselK[1, Sqrt[α^2 - β^2]*δ])PDF[HyperbolicDistribution[α, β, δ, μ], x]
→ (E^(β*(x - μ) - α*Sqrt[δ^2 + (x - μ)^2])*Sqrt[α^2 - β^2])/(2*α*δ*BesselK[1, Sqrt[α^2 - β^2]*δ])Variance[HyperbolicDistribution[α, β, δ, μ]]
→ (δ*BesselK[2, Sqrt[α^2 - β^2]*δ])/(Sqrt[α^2 - β^2]*BesselK[1, Sqrt[α^2 - β^2]*δ]) - (β^2*δ^2*BesselK[2, Sqrt[α^2 - β^2]*δ]^2)/((α^2 - β^2)*BesselK[1, Sqrt[α^2 - β^2]*δ]^2) + (β^2*δ^2*BesselK[3, Sqrt[α^2 - β^2]*δ])/((α^2 - β^2)*BesselK[1, Sqrt[α^2 - β^2]*δ])