Johnson's SU-distribution

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Johnson's S_U

Probability density function
Cumulative distribution function
Parameters	${\displaystyle \gamma ,\xi ,\delta >0,\lambda >0}$ (real)
Support	${\displaystyle -\infty {\text{ to }}+\infty }$
PDF	${\displaystyle {\frac {\delta }{\lambda {\sqrt {2\pi }}}}{\frac {1}{\sqrt {1+\left({\frac {x-\xi }{\lambda }}\right)^{2}}}}e^{-{\frac {1}{2}}\left(\gamma +\delta \sinh ^{-1}\left({\frac {x-\xi }{\lambda }}\right)\right)^{2}}}$
CDF	${\displaystyle \Phi \left(\gamma +\delta \sinh ^{-1}\left({\frac {x-\xi }{\lambda }}\right)\right)}$
Mean	${\displaystyle \xi -\lambda \exp {\frac {\delta ^{-2}}{2}}\sinh \left({\frac {\gamma }{\delta }}\right)}$
Median	${\displaystyle \xi +\lambda \sinh \left(-{\frac {\gamma }{\delta }}\right)}$
Variance	${\displaystyle {\frac {\lambda ^{2}}{2}}(\exp(\delta ^{-2})-1)\left(\exp(\delta ^{-2})\cosh \left({\frac {2\gamma }{\delta }}\right)+1\right)}$
Skewness	${\displaystyle -{\frac {\lambda ^{3}{\sqrt {e^{\delta ^{-2}}}}(e^{\delta ^{-2}}-1)^{2}((e^{\delta ^{-2}})(e^{\delta ^{-2}}+2)\sinh({\frac {3\gamma }{\delta }})+3\sinh({\frac {2\gamma }{\delta }}))}{4(\operatorname {Variance} X)^{1.5}}}}$
Excess kurtosis	${\displaystyle {\frac {\lambda ^{4}(e^{\delta ^{-2}}-1)^{2}(K_{1}+K_{2}+K_{3})}{8(\operatorname {Variance} X)^{2}}}}$ ${\displaystyle K_{1}=\left(e^{\delta ^{-2}}\right)^{2}\left(\left(e^{\delta ^{-2}}\right)^{4}+2\left(e^{\delta ^{-2}}\right)^{3}+3\left(e^{\delta ^{-2}}\right)^{2}-3\right)\cosh \left({\frac {4\gamma }{\delta }}\right)}$ ${\displaystyle K_{2}=4\left(e^{\delta ^{-2}}\right)^{2}\left(\left(e^{\delta ^{-2}}\right)+2\right)\cosh \left({\frac {3\gamma }{\delta }}\right)}$ ${\displaystyle K_{3}=3\left(2\left(e^{\delta ^{-2}}\right)+1\right)}$

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

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

where ${\displaystyle z\sim {\mathcal {N}}(0,1)}$.