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title: "Estimate of the elapsed period t"
canonical: "https://modelassist.epixanalytics.com/space/EA/26574983/Estimate%20of%20the%20elapsed%20period%20t"
format: markdown
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We can estimate the period *t* that has elapsed if we know <span style="color: #222222">λ</span> and the number of events *a* that have occurred in time *t*. The mathematics are exactly the same as the estimate for <span style="color: #222222">λ</span>. The reader may like to verify that, by using a prior of *p(t)* = 1/ *t* we obtain a posterior distribution: *t* = Gamma(1/<span style="color: #222222">λ</span>*,**<span style="color: #222222">α</span>*) 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 <span style="color: #222222">λ</span> = 1/<span style="color: #222222">β</span>. Also, if we can reasonably describe our prior belief of the elapsed period t with a [Gamma](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575231/)(<span style="color: #222222">b</span>,a) distribution, the posterior is given by a Gamma(<span style="color: #222222">b</span>/ (1 + <span style="color: #222222">bλ</span>*),**<span style="color: #222222">α</span>** + a) *distribution.

Note that here we use the parameterization Gamma(b, a) <span style="color: #172b4d"> </span><span style="color: #172b4d">*where*</span><span style="color: #172b4d"> </span>*b*<span style="color: #172b4d"> </span><span style="color: #172b4d">*= Scale and*</span><span style="color: #172b4d"> </span>*a =*<span style="color: #172b4d"> </span><span style="color: #172b4d">*Shape, *</span><span style="color: #172b4d">whereas in other sections of ModelAssist we might report a three-parameter version of the Gamma distribution. </span>

  


  


  


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