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title: "Poisson distribution method of estimating a rate  - not recommended"
canonical: "https://modelassist.epixanalytics.com/space/EA/26575360/Poisson%20distribution%20method%20of%20estimating%20a%20rate%20%20-%20not%20recommended"
format: markdown
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<span style="color: #000000">We assume that each observation is a Poisson random variable that has a rate</span> <span style="color: #000000">. Letting </span><span style="color: #000000">*X*</span><span style="color: #000000"> be the random variable of the observed rate over the observation period (assuming time) t, it will take a distribution given by:</span>

  


     

![image](media://b7906022-6b0e-4510-8b25-9efa3281c8b5)

                                                                 

> Macro (mathinline)

                                                 (1)

  


We observe a events over period t so a/*t* is our one observation from the random variable *X* which is also our [maximum likelihood](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575336), and [unbiased](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26586655)<span style="color: #339966">,</span> estimate for <span style="color: #222222">λ</span>. Switching around Equation 1, with X = <span style="color: #222222">*α*</span>/*t*, we can get an uncertainty distribution for the true value of <span style="color: #222222">λ</span>:

  


![image](media://a9f9ed39-c758-4375-9e81-6feb998e7894)

> Macro (mathinline)

                                                   (2)

                                                                     

  


This exactly equates to the non-parametric and parametric [Bootstrap estimates of a Binomial probability](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575408/). Equation 2 is awkward since it will only allow discrete values for <span style="color: #222222">λ</span> i.e. {0, 1/*t*, 2/*t*, …}, whereas our uncertainty about <span style="color: #222222">λ</span> should really take into account all values greater than zero:

  


![image](media://8b04e6e8-9fcb-4879-9774-873ebeba82c4)

<span style="color: #000000">**Figure 1**</span><span style="color: #000000">: Example of Equation 2 estimate of</span> l <span style="color: #000000">where</span> a <span style="color: #000000">= 2, </span><span style="color: #000000">*t*</span><span style="color: #000000"> = 4</span>

  


<span style="color: #000000">It also makes no sense that</span> <span style="color: #222222">λ</span> <span style="color: #000000">could be either zero (equivalent to the events cannot occur) when we have seen non-zero</span> a<span style="color: #000000">, of course.</span>

  


  


  


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