Slash distribution explained

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

where

\varphi(x) is the probability density function of the standard normal distribution. The quotient is undefined at x = 0, but the discontinuity is removable:\lim _{x\to} f(x) = (\varphi( 0 ))/(2) = (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]

See also

• Scale mixture

References

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