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

*) 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*

*,α**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*

*a**),*distribution.

*α*+*)**a*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.