Slash distribution

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]

Probability density function
Cumulative distribution function
Parameters none
Mean Does not exist
Median 0
Mode 0
Variance Does not exist
Skewness Does not exist
Ex. kurtosis Does not exist
MGF Does not exist

The probability density function (pdf) is

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

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]


  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. "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

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