probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

, the slash distribution is the

probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...

of a standard normal variate divided by an independent standard uniform variate. In other words, if the

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...

''Z'' has a normal distribution with zero mean and unit

variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...

, the random variable ''U'' has a uniform distribution on ,1and ''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 John Wilder Tukey (; June 16, 1915 – July 26, 2000) was an American mathematician and statistician, best known for the development of the fast Fourier Transform (FFT) algorithm and box plot. The Tukey range test, the Tukey lambda distributi ...

in a paper published in 1972. The

probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...

(pdf) is :

f(x) = \frac.

where

\varphi(x)

is the probability density function of the standard normal distribution. The quotient is undefined at ''x'' = 0, but the discontinuity is removable: :

\lim_ f(x) = \frac = \frac

The most common use of the slash distribution is in

simulation A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of models; the model represents the key characteristics or behaviors of the selected system or process, whereas the ...

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 The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fu ...

References

{{ProbDistributions, continuous-infinite Continuous distributions Normal distribution Compound probability distributions Probability distributions with non-finite variance