# Slash distribution

Parameters Probability density function Cumulative distribution function none ${\displaystyle x\in (-\infty ,\infty )}$ ${\displaystyle {\begin{cases}{\frac {\varphi (0)-\varphi (x)}{x^{2}}}&x\neq 0\\{\frac {1}{2{\sqrt {2\pi }}}}&x=0\\\end{cases}}}$ ${\displaystyle {\begin{cases}\Phi (x)-\left[\varphi (0)-\varphi (x)\right]/x&x\neq 0\\1/2&x=0\\\end{cases}}}$ Does not exist 0 0 Does not exist Does not exist Does not exist Does not exist ${\displaystyle {\sqrt {2\pi }}{\Big (}\varphi (t)+t\Phi (t)-\max\{t,0\}{\Big )}}$

In probability theory, the slash distribution is the probability distribution of a standard normal variate divided by an independent standard uniform variate.[1] 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 XZ / 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.[2]

The probability density function (pdf) is

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

where φ(x) is the probability density function of the standard normal distribution.[3] The result is undefined at x = 0, but the discontinuity is removable:

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

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.[3]

## References

1. ^ Davison, Anthony Christopher; Hinkley, D. V. (1997). Bootstrap methods and their application. Cambridge University Press. p. 484. ISBN 978-0-521-57471-6. Retrieved 24 September 2012.
2. ^ Rogers, W. H.; Tukey, J. W. (1972). "Understanding some long-tailed symmetrical distributions". Statistica Neerlandica. 26 (3): 211–226. doi:10.1111/j.1467-9574.1972.tb00191.x.
3. ^ a b "SLAPDF". Statistical Engineering Division, National Institute of Science and Technology. Retrieved 2009-07-02.

This article incorporates public domain material from the National Institute of Standards and Technology website https://www.nist.gov.