Inverse distribution

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 means "is proportional to". It follows that the inverse distribution in this case is of the form which is again a reciprocal 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 For example, if X is uniformly distributed on the interval (0,1), then Y = 1 / X has density and cumulative distribution function when Let X be a t distributed random variate with k degrees of freedom. Then its density function is The density of Y = 1 / X is With k = 1, the distributions of X and 1 / X are identical (X is then Cauchy distributed (0,1)). If k > 1 then the distribution of 1 / X is bimodal.