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

where is the probability density function of the standard normal distribution.^{[3]} The quotient 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]}

## References

**^**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 website https://www.nist.gov.