Slash distribution

From Wikipedia, the free encyclopedia
Jump to: navigation, search
Slash
Probability density function
Slashpdf.svg
Cumulative distribution function
Slashcdf.svg
Parameters none
Support
PDF
CDF
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

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 (pdf) is

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:

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[edit]

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

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. 
  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.