Reciprocal
Probability density function
Cumulative distribution function
Parameters
Support	${\displaystyle [a,b]}$
PDF	${\displaystyle \frac{1}{x\ln \frac{b}{a}}}$
CDF	${\displaystyle \frac{\ln \frac{x}{a}}{\ln \frac{b}{a}}}$
Mean	${\displaystyle \frac{b-a}{\ln \frac{b}{a}}}$
Median	${\displaystyle \sqrt{ab}}$
Mode	${\displaystyle a}$
Variance	${\displaystyle \frac{b^2-a^2}{2\ln \frac{b}{a}}-{\left(\frac{b-a}{\ln \frac{b}{a}}\right)}^2}$
Entropy	${\displaystyle \ln \left(\ln \left(\frac{b}{a}\right)\right)+\frac{\ln {\left(b\right)}^2-\ln {\left(a\right)}^2}{2\ln \left(\frac{b}{a}\right)}}$
MGF	${\displaystyle \frac{\mathrm{E}\mathrm{i}(bt)-\mathrm{E}\mathrm{i}(at)}{\ln \left(b\right)-\ln \left(a\right)}}$
CF	${\displaystyle \frac{\mathrm{E}\mathrm{i}(ibt)-\mathrm{E}\mathrm{i}(iat)}{\ln \left(b\right)-\ln \left(a\right)}}$

Inprobability andstatistics, thereciprocal distribution, also known as thelog-uniform distribution, is acontinuous probability distribution. It is characterised by itsprobability density function, within the support of the distribution, being proportional to thereciprocal of the variable.

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

Theprobability density function (pdf) of the reciprocal distribution is

${\displaystyle f(x;a,b)=\frac{1}{x[\ln (b)-\ln (a)]}\text{ for }a\le x\le b\text{ and }a>0.}$

Here,${\displaystyle a}$ and${\displaystyle b}$ are the parameters of the distribution, which are the lower and upper bounds of thesupport, and${\displaystyle \ln }$ is thenatural log. Thecumulative distribution function is

${\displaystyle F(x;a,b)=\frac{\ln (x)-\ln (a)}{\ln (b)-\ln (a)}\text{ for }a\le x\le b.}$

A positive random variableX is log-uniformly distributed if the logarithm ofX is uniform distributed,

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

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

The reciprocal distribution is of considerable importance innumerical analysis, because acomputer’s arithmetic operations, in particular, repeated multiplications and/or divisions, transformmantissas with initial arbitrary distributions into the reciprocal distribution as a limiting distribution.^[1]

• Continuous distributions

• Articles with short description
• Short description matches Wikidata