# Johnson SU distribution

Parameters $\gamma, \xi, \delta > 0, \lambda > 0$ (real) $-\infty \text{ to } +\infty$ $\frac{\delta}{\lambda\sqrt{2\pi}}\frac{1}{\sqrt{1 + (\frac{x-\xi}{\lambda})^{2}}}e^{-\frac{1}{2}(\gamma+\delta \sinh^{-1}(\frac{x-\xi}{\lambda}))^{2}}$ $\Phi (\gamma + \delta \sinh^{-1}(\frac{x-\xi}{\lambda}))$ $\xi - \lambda \exp{\frac{\delta^{-2}}{2}} \sinh(\frac{\gamma}{\delta})$ $\frac{\lambda^2}{2}(\exp{(\delta^{-2})}-1)(\exp{(\delta^{-2})}\cosh{(\frac{2\gamma}{\delta})}+1)$

The Johnson $S_U$ distribution is a four-parameter family of probability distributions first investigated by N. L. Johnson in 1949.[1] Johnson proposed it as a transformation of the normal distribution:[2]

$z=\gamma+\delta \sinh^{-1} (\frac{x-\xi}{\lambda})$

Where $z \sim \mathcal{N}(0,1)$

## Generation of random variables

Let U be a random variable that is uniformly distributed on the unit interval [0, 1]. Johnson $S_U$ random variables can be generated from U as follows:

$x = \lambda \sinh\left( \frac{ \Phi^{ -1 }( U ) - \gamma }{ \delta } \right) + \xi$

where Φ is the cumulative distribution function of the normal distribution.

## References

1. ^ Johnson, N. L. (1949) Systems of frequency curves generated by methods of translation. Biometrika 36: 149–176 JSTOR 2332539
2. ^ N.L. Johnson, Table to facilitate fitting SU frequency curves, Biometrika 1949