Slash distribution

name =Slash| type =density| pdf_image =[[File:Slashpdf.svg|275px|center]] | cdf_image =[[File:Slashcdf.svg|275px|center]]| parameters =none| support =x\in(-\infty,\infty)| pdf =\begin{cases} \frac{\varphi(0) - \varphi(x)}{x^2} & x \ne 0 \ \frac{1}{2\sqrt{2\pi}} & x = 0 \ \end{cases} | cdf =\begin{cases} \Phi(x) - \left[ \varphi(0) - \varphi(x) \right] / x & x \ne 0 \ 1 / 2 & x = 0 \ \end{cases} | mean =Does not exist| median =0| mode =0| variance =Does not exist| skewness =Does not exist| kurtosis =Does not exist| entropy =| mgf =Does not exist | char = \sqrt{2\pi}\Big(\varphi(t)+t\Phi(t)-\max{t,0}\Big) | 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

: f(x) = \frac{\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 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.