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We can estimate the period t that has elapsed if we know λ and the number of events a that have occurred in time t. The mathematics are exactly the same as the estimate for λ. The reader may like to verify that, by using a prior of p(t) = 1/ t we obtain a posterior distribution: t = Gamma(1/λ,α) which is the same result we would obtain if we were trying to predict forward (i.e. determine a distribution of variability of) the time required to observe a events given λ = 1/β. Also, if we can reasonably describe our prior belief of the elapsed period t with a Gamma(b,a) distribution, the posterior is given by a Gamma(b/ (1 + bλ),α + a) distribution.

Note that here we use the parameterization Gamma(b, a)  where b = Scale and a = Shape, whereas in other sections of ModelAssist we might report a three-parameter version of the Gamma distribution. 

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