Reciprocal distribution

In probability and statistics, the reciprocal distribution, also known as the log-uniform distribution, is a continuous probability distribution. It is characterised by its probability density function, within the support of the distribution, being proportional to the reciprocal of the variable.

Reciprocal
Probability density function
Cumulative distribution function
Parameters	$0<a<b,a,b\in \mathbb {R}$
Support	$[a,b]$
PDF	${\frac {1}{x\ln {\frac {b}{a}}}}$
CDF	$\log _{\frac {b}{a}}{\frac {x}{a}}$
Mean	${\frac {b-a}{\ln {\frac {b}{a}}}}$
Variance	${\frac {b^{2}-a^{2}}{2\ln {\frac {b}{a}}}}-\left({\frac {b-a}{\ln {\frac {b}{a}}}}\right)^{2}$

The reciprocal distribution is an example of an inverse distribution, and the reciprocal (inverse) of a random variable with a reciprocal distribution itself has a reciprocal distribution.

Definition

The probability density function (pdf) of the reciprocal distribution is

f(x;a,b)={\frac {1}{x[\log _{e}(b)-\log _{e}(a)]}}\quad {\text{ for }}a\leq x\leq b{\text{ and }}a>0.

Here, $a$ and $b$ are the parameters of the distribution, which are the lower and upper bounds of the support, and $\log _{e}$ is the natural log function (the logarithm to base e). The cumulative distribution function is

F(x;a,b)={\frac {\log _{e}(x)-\log _{e}(a)}{\log _{e}(b)-\log _{e}(a)}}\quad {\text{ for }}a\leq x\leq b.

Characterization

Relationship to Log-Uniform

Histogram and log-histogram of random deviates from the reciprocal distribution

A positive random variable X is log-uniformly distributed if the logarithm of X is uniform distributed,

\ln(X)\sim {\mathcal {U}}(\ln(a),\ln(b)).

This relationship is true regardless of the base of the logarithmic or exponential function. If $\log _{a}(Y)$ is uniform distributed, then so is $\log _{b}(Y)$ , for any two positive numbers $a,b\neq 1$ . Likewise, if $e^{X}$ is log-uniform distributed, then so is $a^{X}$ , where $0<a\neq 1$ .

Applications

The reciprocal distribution is of considerable importance in numerical analysis as a computer’s arithmetic operations transform mantissas with initial arbitrary distributions to the reciprocal distribution as a limiting distribution.[1]

References

Hamming R. W. (1970) "On the distribution of numbers", The Bell System Technical Journal 49(8) 1609–1625

Probability distributions (List)
Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher soliton discrete uniform Zipf Zipf–Mandelbrot
Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Flory–Schulz Gauss–Kuzmin geometric logarithmic negative binomial Panjer parabolic fractal Poisson Skellam Yule–Simon zeta
Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular continuous Bernoulli Irwin–Hall Kumaraswamy logit-normal noncentral beta PERT 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 Fréchet gamma gamma/Gompertz generalized gamma 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 noncentral F 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 chi-squared generalized extreme value generalized Pareto Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace 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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