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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.
Relation to original 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:
where means "is proportional to". It follows that the inverse distribution in this case is of the form
which is again a reciprocal distribution.
Inverse uniform distribution
If the original random variable X is uniformly distributed on the interval (a,b), where a>0, then the reciprocal variable Y = 1 / X has the reciprocal distribution which takes values in the range (b−1 ,a−1), and the probability density function in this range is
and is zero elsewhere.
The cumulative distribution function of the reciprocal, within the same range, is
Inverse t distribution
The density of Y = 1 / X is
Reciprocal normal distribution
Inverse Cauchy distribution
If X is a Cauchy distributed (μ, σ) random variable, then 1 / X is a Cauchy ( μ / C, σ / C ) random variable where C = μ2 + σ2.
Inverse F distribution
If X is an F(ν1, ν2 ) distributed random variable then 1 / X is an F(ν2, ν1 ) random variable.
Other inverse distributions
Inverse distributions are widely used as prior distributions in Bayesian inference for scale parameters.
- Hamming R. W. (1970) "On the distribution of numbers", The Bell System Technical Journal 49(8) 1609–1625
- Johnson, Norman L.; Kotz, Samuel; Balakrishnan, Narayanaswamy (1994). Continuous Univariate Distributions, Volume 1. Wiley. p. 171. ISBN 0-471-58495-9. (this is a special case of the generalized inverse normal distribution treated)