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2012-01-01T03:17:57Z

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{{Probability distribution|
name =Slash|
type =density|
pdf_image =[[File:Slashpdf.svg|275px|center]] |
cdf_image =[[File:Slashcdf.svg|275px|center]]|
parameters =none|
support =<math>x\in(-\infty,\infty)</math>|
pdf =<math>\frac{\varphi(0) - \varphi(x)}{x^2} </math> |
cdf =<math>\begin{cases}
\Phi(x) - \left[ \varphi(0) - \varphi(x) \right] / x & x \ne 0 \\
1 / 2 & x = 0 \\
\end{cases}</math> |
mean =Does not exist|
median =0|
mode =0|
variance =Does not exist|
skewness =Does not exist|
kurtosis =Does not exist|
entropy =|
mgf =Does not exist |
char = <math>\sqrt{2\pi}\Big(\varphi(t)+t\Phi(t)-\max\{t,0\}\Big)</math> |
}}
In [[probability theory]], the '''slash distribution''' is the [[probability distribution]] of a standard [[normal distribution|normal]] variate divided by an independent [[uniform distribution (continuous)#Standard uniform|standard uniform]] variate.<ref>{{cite book|last1=Davison|first1=Anthony Christopher|last2=Hinkley|first2=D. V.|authorlink2=David V. Hinkley|title=Bootstrap methods and their application |publisher=Cambridge University Press|url=http://www.cambridge.org/us/knowledge/isbn/item1154176/?site_locale=en_US |date=1997|isbn=978-0-521-57471-6|page=484|accessdate=24 September 2012}}</ref> In other words, if the [[random variable]] ''Z'' has a normal distribution with zero mean and unit [[variance]], the random variable ''U'' has a uniform distribution on [0,1] and ''Z'' and ''U'' are [[statistically independent]], then the random variable ''X'' = ''Z'' / ''U'' has a slash distribution. The slash distribution is an example of a [[ratio distribution]]. The distribution was named by William H. Rogers and [[John Tukey]] in a paper published in 1972.<ref>{{cite doi|10.1111/j.1467-9574.1972.tb00191.x}}</ref>

The [[probability density function]] is

:<math> f(x) = \frac{\varphi(0) - \varphi(x)}{x^2}.</math>

where φ(''x'') is the probability density function of the standard normal distribution.<ref name=nist /> The result is undefined at ''x'' = 0, but the [[removable discontinuity|discontinuity is removable]]:

: <math> \lim_{x\to 0} f(x) = \frac{\varphi(0)}{2} = \frac{1}{2\sqrt{2\pi}} </math>

The most common use of the slash distribution is in [[simulation]] studies. It is a useful distribution in this context because it has [[heavy tail|heavier tails]] than a normal distribution, but it is not as [[pathological (mathematics)|pathological]] as the [[Cauchy distribution]].<ref name=nist>{{cite web|url=http://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/slapdf.htm|title=SLAPDF|publisher=Statistical Engineering Division, National Institute of Science and Technology|accessdate=2009-07-02}}</ref>

==References==
<references/>

{{NIST-PD}}

{{ProbDistributions|continuous-infinite}}

[[Category:Continuous distributions]]
[[Category:Normal distribution]]
[[Category:Probability distributions with non-finite variance]]
[[Category:Probability distributions]]

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