Slash distribution

In probability theory, the slash distribution is the probability distribution of a standard normal variate divided by an independent standard uniform variate. 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.

The probability density function (pdf) is

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

where ${\displaystyle \varphi (x)}$is the probability density function of the standard normal distribution. The quotient 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.

• Scale mixture

 This article incorporates public domain material from the National Institute of Standards and Technology