Slash distribution

Slash

Probability density function
Cumulative distribution function
Parameters	none
Support	$x\in(-\infty,\infty)$
PDF	$\frac{\varphi(0) - \varphi(x)}{x^2}$
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
Ex. kurtosis	Does not exist
MGF	Does not exist
CF	$\sqrt{2\pi}\Big(\varphi(t)+t\Phi(t)-\max\{t,0\}\Big)$

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 φ(x) is the probability density function of the standard normal distribution. The result 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.

Differential equation

The pdf of the slash distribution is a solution of the following differential equation:

$\left\{\begin{array}{l} 2 \sqrt{\pi} x f'(x)+2 \sqrt{\pi} \left(x^2+2\right) f(x)-\sqrt{2}=0, \\[12pt] f(1)=\frac{1}{\sqrt{2 \pi}}-\frac{1}{\sqrt{2 e\pi}} \end{array}\right\}$

References

 This article incorporates public domain material from websites or documents of the National Institute of Standards and Technology.