< List of probability distributions < Slash distribution
The slash distribution (also called the slash normal or ordinary slash) is a type of a ratio distribution. It is the distribution of the ratio of a normal random variable to an independent uniform random variable (i.e., N(0, 1)/U(0, 1)) with symmetric probability density .
The distribution is known for its extreme outliers and heavier than normal tails (i.e., greater kurtosis) . It is often used as a challenging distribution for statistical procedures such as investigating how variables perform under extreme conditions , including robustness studies (e.g., [4, 5]) and simulation studies; it is not as pathological as the Cauchy distribution, which is used for the same purposes.
The slash distribution was given its name by William H. Rogers and John Tukey in a paper published in their 1972 paper titled Understanding some long-tailed symmetrical distributions . The authors introduced the distribution with the following stochastic representation:
Y = μ + σ (Z/U1/q)
- q > 0 = shape parameter (controls tail thickness and kurtosis)
- μ ∈ ℝ = location parameter
- σ > 0 = scale parameter.
Many variants exist, including the slash (student) t distribution and the skew-slash (student) t distribution which includes degrees of freedom as an extra parameter .
Slash distribution properties
The slash distribution probability density function (PDF) can be expressed in terms of the standard normal density φ(x) :
The quotient has a discontinuity at zero and is therefore undefined at that point. A workaround is to remove the discontinuity with
The slash cumulative distribution function (CDF) is computed by numerically integrating the slash probability density function 
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