Reciprocal distribution

Reciprocal

Parameters	${\displaystyle 0<a<b,a,b\in \mathbb {R} }$
Support	${\displaystyle [a,b]}$
PDF	${\displaystyle {\frac {1}{x[\log _{e}(b)-\log _{e}(a)]}}}$
CDF	${\displaystyle {\frac {\log _{e}(x)-\log _{e}(a)}{\log _{e}(b)-\log _{e}(a)}}}$
Mean	${\displaystyle {\frac {b-a}{\log _{e}(b)-\log _{e}(a)}}}$

In probability and statistics, the reciprocal 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.

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.

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

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

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