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.[1] 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.[2]

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.[3] 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.[3]

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

  1. ↑ Davison, Anthony Christopher; Hinkley, D. V. (1997). Bootstrap methods and their application. Cambridge University Press. p. 484. ISBN 978-0-521-57471-6. Retrieved 24 September 2012.
  2. ↑ Rogers, W. H.; Tukey, J. W. (1972). "Understanding some long-tailed symmetrical distributions". Statistica Neerlandica 26 (3): 211–226. doi:10.1111/j.1467-9574.1972.tb00191.x.
  3. ↑ 3.0 3.1 "SLAPDF". Statistical Engineering Division, National Institute of Science and Technology. Retrieved 2009-07-02.

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