# Johnson SU distribution

Parameters γ, ξ, σ > 0, λ > 0 (real) $-\infty \text{ to } +\infty$ $\Phi (\gamma + \sigma \sinh^{-1}z)$ $\chi - \lambda \exp{\frac{\sigma^{-2}}{2}} \sinh(\frac{\gamma}{\sigma})$ $\frac{\lambda^2}{2}(\exp{(\sigma^{-2})}-1)(\exp{(\sigma^{-2})}\cosh^{2}{(\frac{2\gamma}{\sigma})}+1)$

The Johnson $S_U$ distribution is a four-parameter family of probability distributions first investigated by N. L. Johnson in 1949.[1] It is closely related to the normal distribution.

## 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:

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

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