Slash distribution

Slash
	Probability density function
	Cumulative distribution function
Parameters	none
Support
PDF
CDF
Mean	Does not exist
Median	0
Mode	0
Variance	Does not exist
Skewness	Does not exist
Excess kurtosis	Does not exist
MGF	Does not exist
CF

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 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.^[2]

The probability density function (pdf) is

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

where $\varphi (x)$ is the probability density function of the standard normal distribution.^[3] The quotient is undefined at x = 0, but the discontinuity is removable:

\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[edit]

^ 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.
^ 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.
^ ^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

[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.

[nist-3] "SLAPDF". Statistical Engineering Division, National Institute of Science and Technology. Retrieved 2009-07-02.

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