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The equation, parameters and bounds of the density function are: 0 ⎧ 2 ⎪⎪ ⎛ x −m+ r ⎞ ⎞ 1⎛ 2 ⎟⎟ − ⎜ γ + η ln ⎜ f (x η, γ, r, m ) = ⎨ r ⎟ ⎜ ⎜ m+ −x ⎟ 2 ηr 2 ⎝ ⎠⎠ e ⎝ ⎪ r r ⎪⎩ 2π (x − m + 2 )(m + 2 − x ) Parameters: Location: Scale: Shape: Shape: Bounds: Lower bound = m – r/2 Upper bound = m + r/2 Moments - SB: expression. m r γ δ x ≤ m − 2r , x ≥ m + m − 2r < x < m + r 2 r 2 -∞ < m < ∞ r>0 -∞ < γ < ∞ η>0 The moments of the Johnson SB distribution do not have a simple Density Function - SU: The density function of the Johnson SU distribution is shown below: f ε = 0, λ = 1, γ = 1 0.7 η=2 0.6 0.5 η=1 0.4 0.3 η = 0.5 0.2 0.1 x -3 -2 -1 1 Distributions 2 3 79
