Support − ∞ to + ∞ {displaystyle -infty { ext{ to }}+infty } PDF δ λ 2 π 1 1 + ( x − ξ λ ) 2 e − 1 2 ( γ + δ sinh − 1 ( x − ξ λ ) ) 2 {displaystyle { rac {delta }{lambda {sqrt {2pi }}}}{ rac {1}{sqrt {1+left({ rac {x-xi }{lambda }}ight)^{2}}}}e^{-{ rac {1}{2}}left(gamma +delta sinh ^{-1}left({ rac {x-xi }{lambda }}ight)ight)^{2}}} CDF Φ ( γ + δ sinh − 1 ( x − ξ λ ) ) {displaystyle Phi left(gamma +delta sinh ^{-1}left({ rac {x-xi }{lambda }}ight)ight)} Mean ξ − λ exp δ − 2 2 sinh ( γ δ ) {displaystyle xi -lambda exp { rac {delta ^{-2}}{2}}sinh left({ rac {gamma }{delta }}ight)} Variance λ 2 2 ( exp ( δ − 2 ) − 1 ) ( exp ( δ − 2 ) cosh ( 2 γ δ ) + 1 ) {displaystyle { rac {lambda ^{2}}{2}}(exp(delta ^{-2})-1)left(exp(delta ^{-2})cosh left({ rac {2gamma }{delta }}ight)+1ight)} | ||
The Johnson's SU-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:
Contents
where
Generation of random variables
Let U be a random variable that is uniformly distributed on the unit interval [0, 1]. Johnson's SU random variables can be generated from U as follows:
where Φ is the cumulative distribution function of the normal distribution.
Additional reading
References
Johnson's SU-distribution Wikipedia(Text) CC BY-SA