Parameters γ
,
ξ
,
δ
>
0
,
λ
>
0
{displaystyle gamma ,xi ,delta >0,lambda >0}
(real) 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)+1
ight)} | ||
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