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	${\begin{cases}\Phi (x)-\left[\varphi (0)-\varphi (x)\right]/x&x\neq 0\\1/2&x=0\\\end{cases}}$
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

↑ 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.
↑ 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.
1 2 "SLAPDF". Statistical Engineering Division, National Institute of Science and Technology. Retrieved 2009-07-02.

This article incorporates public domain material from the National Institute of Standards and Technology website http://www.nist.gov.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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