The Johnson's S<sub>U</sub>-distribution is a four-parameter family of probability distributions first investigated by N. L. Johnson in 1949. Johnson proposed it as a transformation of the normal distribution:
where .
Let U be a random variable that is uniformly distributed on the unit interval [0, 1]. Johnson's S<sub>U</sub> random variables can be generated from U as follows:
where æ is the cumulative distribution function of the normal distribution.
N. L. Johnson firstly proposes the transformation :
where .
Johnson's S<sub>B</sub> random variables can be generated from U as follows:
The S<sub>B</sub>-distribution is convenient to Platykurtic distributions (Kurtosis). To simulate S<sub>U</sub>, sample of code for its density and cumulative distribution function is available here
Johnson's -distribution has been used successfully to model asset returns for portfolio management. This comes as a superior alternative to using the Normal distribution to model asset returns. An R package, JSUparameters, was developed in 2021 to aid in the estimation of the parameters of the best-fitting Johnson's -distribution for a given dataset. Johnson distributions are also sometimes used in option pricing, so as to accommodate an observed volatility smile; see Johnson binomial tree.
An alternative to the Johnson system of distributions is the quantile-parameterized distributions (QPDs). QPDs can provide greater shape flexibility than the Johnson system. Instead of fitting moments, QPDs are typically fit to empirical CDF data with linear least squares.
Johnson's -distribution is also used in the modelling of the invariant mass of some heavy mesons in the field of B-physics.