slash distribution

{{Short description|Concept in probability theory}}

{{Probability distribution|

name =Slash|

type =density|

pdf_image =File:Slashpdf.svg |

cdf_image =File:Slashcdf.svg|

parameters =none|

support =$x\backslash in(-\backslash infty,\backslash infty)$|

pdf =$\backslash begin\{cases\}$

\frac{\varphi(0) - \varphi(x)}{x^2} & x \ne 0 \\

\frac{1}{2\sqrt{2\pi}} & x = 0 \\

\end{cases} |

cdf =$\backslash begin\{cases\}$

\Phi(x) - \left[ \varphi(0) - \varphi(x) \right] / x & x \ne 0 \\

1 / 2 & x = 0 \\

\end{cases} |

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 = $\backslash sqrt\{2\backslash pi\}\backslash Big(\backslash varphi(t)+t\backslash Phi(t)-\backslash max\backslash \{t,0\backslash \}\backslash Big)$ |

}}

In probability theory, the slash distribution is the probability distribution of a standard normal variate divided by an independent standard uniform variate.{{cite book|last1=Davison|first1=Anthony Christopher|last2=Hinkley|first2=D. V.|author-link2=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|access-date=24 September 2012}} 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.{{Cite journal| last1 = Rogers | first1 = W. H.| last2 = Tukey | first2 = J. W.| author-link2 = John Tukey| title = Understanding some long-tailed symmetrical distributions| journal = Statistica Neerlandica | volume = 26| issue = 3 | pages = 211–226 | year = 1972 | doi = 10.1111/j.1467-9574.1972.tb00191.x}}

The probability density function (pdf) is

:$f(x)\; =\; \backslash frac\{\backslash varphi(0)\; -\; \backslash varphi(x)\}\{x^2\}.$

where $\backslash varphi(x)$ is the probability density function of the standard normal distribution. The quotient is undefined at x = 0, but the discontinuity is removable:

: $\backslash lim\_\{x\backslash to\; 0\}\; f(x)\; =\; \backslash frac\{\backslash varphi(0)\}\{2\}\; =\; \backslash frac\{1\}\{2\backslash sqrt\{2\backslash pi\}\}$

The most common use of the slash distribution is in simulation studies. It is a useful distribution in this context because it has heavier tails than a normal distribution, but it is not as pathological as the Cauchy distribution.{{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|access-date=2009-07-02}}