Slash distribution

Slash

Probability density function
Cumulative distribution function Error creating thumbnail: Unable to save thumbnail to destination
Parameters	none
Support	$x\in (-\mathrm{\infty },\mathrm{\infty })$
PDF	$\{\begin{array}{ll}\frac{\phi (0)-\phi (x)}{x^2}& x\ne 0\\ \frac{1}{2\sqrt{2\pi }}& x=0\\ \end{array}$
CDF	$\{\begin{array}{ll}\mathrm{\Phi }(x)-[\phi (0)-\phi (x)]/x& x\ne 0\\ 1/2& x=0\\ \end{array}$
Mean	Does not exist
Median	0
Mode	0
Variance	Does not exist
Skewness	Does not exist
Kurtosis	Does not exist
MGF	Does not exist
CF	$\sqrt{2\pi }(\phi (t)+t\mathrm{\Phi }(t)-\max \{t,0\})$

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

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

where $\phi (x)$ is the probability density function of the standard normal distribution. The quotient is undefined at x = 0, but the discontinuity is removable:

$\underset{x\to 0}{\lim }f(x)=\frac{\phi (0)}{2}=\frac{1}{2\sqrt{2\pi }}$

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.

References

 This article incorporates public domain material from the National Institute of Standards and Technology website https://www.nist.gov.