Inverse distribution

Inverse uniform distribution

Parameters	${\displaystyle 0<a<b,\quad a,b\in \mathbb {R} }$
Support	${\displaystyle [b^{-1},a^{-1}]}$
PDF	${\displaystyle y^{-2}{\frac {1}{b-a}}}$
CDF	${\displaystyle {\frac {b-y^{-1}}{b-a}}}$
Mean	${\displaystyle {\frac {\ln({\frac {1}{a}})-\ln({\frac {1}{b}})}{b-a}}}$
Median	${\displaystyle {\frac {2}{a+b}}}$
Variance	${\displaystyle {\frac {1}{a\cdot b}}-\left({\frac {\ln({\frac {1}{a}})-\ln({\frac {1}{b}})}{b-a}}\right)^{2}}$

In probability theory and statistics, an inverse distribution is the distribution of the reciprocal of a random variable. Inverse distributions arise in particular in the Bayesian context of prior distributions and posterior distributions for scale parameters. In the algebra of random variables, inverse distributions are special cases of the class of ratio distributions, in which the numerator random variable has a degenerate distribution.

In general, given the probability distribution of a random variable X with strictly positive support, it is possible to find the distribution of the reciprocal, Y = 1 / X. If the distribution of X is continuous with density function f(x) and cumulative distribution function F(x), then the cumulative distribution function, G(y), of the reciprocal is found by noting that

Then the density function of Y is found as the derivative of the cumulative distribution function:

The reciprocal distribution has a density function of the form.

where ${\displaystyle \propto \!\,}$ means "is proportional to". It follows that the inverse distribution in this case is of the form

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