Slash distribution

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Slash
Probability density function
Slashpdf.svg
Cumulative distribution function
Slashcdf.svg
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 XZ / 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 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


\left\{2 \sqrt{\pi } x f'(x)+2 \sqrt{\pi } \left(x^2+2\right)
   f(x)-\sqrt{2}=0,f(1)=\frac{1}{\sqrt{2 \pi }}-\frac{1}{\sqrt{2 e \pi
   }}\right\}

References[edit]

  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.  edit
  3. ^ a b "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.