Slash
Probability density function
Cumulative distribution function
Parameters	none
Support	${\displaystyle x\in (-\mathrm{\infty },\mathrm{\infty })}$
PDF
CDF
Mean	Does not exist
Median	0
Mode	0
Variance	Does not exist
Skewness	Does not exist
Excess kurtosis	Does not exist
MGF	Does not exist
CF	${\displaystyle \sqrt{2\pi }(\phi (t)+t\mathrm{\Phi }(t)-max\{t,0\})}$

Inprobability theory, theslash distribution is theprobability distribution of a standardnormal variate divided by an independentstandard uniform variate.^[1] In other words, if therandom variableZ has a normal distribution with zero mean and unitvariance, the random variableU has a uniform distribution on [0,1] andZ andU arestatistically independent, then the random variableX = Z / U has a slash distribution. The slash distribution is an example of aratio distribution. The distribution was named by William H. Rogers andJohn Tukey in a paper published in 1972.^[2]

Theprobability density function (pdf) is

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

where${\displaystyle \phi (x)}$ is the probability density function of the standard normal distribution.^[3] The quotient is undefined atx = 0, but thediscontinuity is removable:

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

The most common use of the slash distribution is insimulation studies. It is a useful distribution in this context because it hasheavier tails than a normal distribution, but it is not aspathological as theCauchy distribution.^[3]

• Scale mixture

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

• Continuous distributions
• Normal distribution
• Compound probability distributions
• Probability distributions with non-finite variance

• Articles with short description
• Short description matches Wikidata
• Wikipedia articles incorporating text from the National Institute of Standards and Technology