en>Metadox: fixed typo

2013-04-18T23:33:24Z

fixed typo

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{{DISPLAYTITLE:Johnson ''S<sub>U</sub>'' distribution}}
{{Expert-subject |Statistics |2= |talk= |reason=completion to reasonable standard for probability distributions|date=November 2012 }}

{{Probability distribution
| name = Johnson <math>S_U</math>
| type = continuous
| pdf_image =
| cdf_image =
| notation =
| parameters = ''γ'', ''ξ'', σ > 0, ''λ'' > 0 ([[real number|real]])
| support = <math> -\infty \text{ to } +\infty </math>
| cdf = <math>\Phi (\gamma + \sigma \sinh^{-1}z)</math>
| mean = <math>\chi - \lambda \exp{\frac{\sigma^{-2}}{2}} \sinh(\frac{\gamma}{\sigma})</math>
| median =
| mode =
| variance = <math>\frac{\lambda^2}{2}(\exp{(\sigma^{-2})}-1)(\exp{(\sigma^{-2})}\cosh^{2}{(\frac{2\gamma}{\sigma})}+1)</math>
| skewness =
| kurtosis =
| entropy =
| pgf =
| mgf =
| char =
}}

The '''Johnson <math>S_U</math> distribution''' is a four-parameter family of [[probability distribution]]s first investigated by [[Norman Lloyd Johnson|N. L. Johnson]] in 1949.<ref name=Johnson1949>[[Norman Lloyd Johnson|Johnson, N. L.]] (1949) Systems of frequency curves generated by methods of translation. ''[[Biometrika]]'' '''36''': 149–176 {{JSTOR|2332539}}</ref> It is closely related to the [[normal distribution]].

==Generation of random variables==

Let ''U'' be a [[random variable]] that is [[uniform distribution (continuous)|uniformly distributed]] on the unit interval [0, 1]. Johnson <math>S_U</math> random variables can be generated from ''U'' as follows:

: <math> x = \lambda \sinh\left( \frac{ 1 }{ \sigma } \Phi^{ -1 }( U ) - \gamma \right) + \chi </math>

where Φ is the [[cumulative distribution function]] of the [[normal distribution]].

==References==
{{reflist}}

==Additional reading==
* {{cite doi|10.1093/biomet/asp053}}( [http://oro.open.ac.uk/22510 Preprint])

{{ProbDistributions|continuous-infinite}}
[[Category:Continuous distributions]]
[[Category:Probability distributions]]

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en>Metadox: fixed typo