Estimate of the elapsed period t

For a Poisson process, we can estimate the period t that has elapsed if we know l and the number of events a that have occurred in time t. The maths turns out to be exactly the same as the estimate for l. The reader may like to verify that, by using a prior of p(t) = 1/ t we obtain a posterior distribution: t = Gamma(a,1/l) 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 l = 1/b. Also, if we can reasonably describe our prior belief with a Gamma(a,b) distribution, the posterior is given by a Gamma(a + a, b/ (1 + b l)) distribution.

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