---
title: "Comparison of classical estimates of Poisson lambda and beta"
canonical: "https://modelassist.epixanalytics.com/space/EA/26575364/Comparison%20of%20classical%20estimates%20of%20Poisson%20lambda%20and%20beta"
format: markdown
---
We have already given classical statistics techniques for determining confidence distributions for [λ](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575359) and for [β](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575363/) from a [Poisson process](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575359). However, <span style="color: #222222">λ </span>and <span style="color: #222222">β</span> are just the reciprocal of each other:

  


<span style="color: #222222">β</span> = 1/<span style="color: #222222">λ</span>

  


Thus, an interesting question is: Does our estimate vary depending on the method used?

  


Let's assume that the Poisson process is about events occurring over a period of time t. The estimate for <span style="color: #222222">β</span> considers the time between consecutive events ({t<sub>i</sub>}, so built into that analysis is the assumption that the last of the <span style="color: #222222">α</span> events was observed at a moment equal to the sum of all the individual times (t = SUM({t<sub>i</sub>}).

  


The estimate for <span style="color: #222222">λ</span> however considers a total amount of time t and the number of events within that time <span style="color: #222222">α</span>. There is no assumption that the last event occurred at time t, so t in this analysis in all probability included some amount of time, from the last event to the finish, when nothing happened.

  


Thus <span style="color: #222222">α</span>/t, the observed rate will be higher in the <span style="color: #222222">β</span> estimating scenario than in the <span style="color: #222222">λ</span> estimating scenario. That means that the <span style="color: #222222">β</span> method will systematically estimate a lower value for <span style="color: #222222">λ</span> (=1/<span style="color: #222222">β</span>) than the <span style="color: #222222">λ</span> estimating method. Of course, the difference becomes progressively smaller the larger the number of events <span style="color: #222222">α</span>.

  


  


  


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