---
title: "Cumulative confidence construction estimate for the Poisson intensity"
canonical: "https://modelassist.epixanalytics.com/space/EA/26575362/Cumulative%20confidence%20construction%20estimate%20for%20the%20Poisson%20intensity"
format: markdown
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<span style="color: #000000">The confidence we have that the true value of the intensity is less than or equal to</span> <span style="color: #000000">λ</span> <span style="color: #000000">is:</span>

 

<span style="color: #000000">P(X></span><span style="color: #222222">α</span><span style="color: #000000">; λ) + ½ P(X=</span><span style="color: #222222">α</span><span style="color: #000000">; λ)</span>

 

<span style="color: #000000">where X is the random variable of the number of events one could observe in a period with an expected rate of</span> <span style="color: #000000">λ. This is saying that the greater the true value of the intensity</span> <span style="color: #000000">λ, the more confident we would be in observing a particular number of events</span> <span style="color: #222222">α</span><span style="color: #000000">, or more. Translated into Excel formula, the confidence we have that the true Poisson intensity is less than any specific tested value</span> <span style="color: #000000">λ</span> <span style="color: #000000">is given by:</span>

 

<span style="color: #000000">                =1-POISSON(</span><span style="color: #222222">α</span><span style="color: #000000">,λ,1)+0.5*POISSON(</span><span style="color: #222222">α</span><span style="color: #000000">,λ,0)</span>

 

<span style="color: #000000">We can use this to construct a cumulative confidence distribution by testing a range of values of</span> <span style="color: #000000">λ:</span>

 

![image](media://25fd85d5-b176-453b-86a1-aeaec8ca348d)

 

<span style="color: #000000">**Figure 1**</span><span style="color: #000000">: Cumulative distributions of estimate of</span> <span style="color: #000000">λ</span> <span style="color: #000000">for varying number of observations</span> <span style="color: #222222">α</span><span style="color: #000000"> </span>

 

 

The points used to construct the above plot can be fed into a [Custom Distribution](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575279/) to construct a [Cumulative Distribution](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575222/) so that you can generate values from it. The following model show how: Poisson Confidence Construction;

 

Looking at Figure 1, you will see that the cumulative distribution for <span style="color: #222222">α</span> = 0 starts at 0.5. That means the distribution is assigning 50% confidence to <span style="color: #000000">λ</span> = 0, and the remaining 50% confidence to all other values of <span style="color: #000000">λ</span>. That is equivalent to saying that when there have been no observations, we are equally confident that no such stochastic process exists. The Bayesian equivalent would be to assign a prior distribution with 1/2 confidence assigned to <span style="color: #000000">λ</span>=0, and 1/2 confidence distributed over (0,∞). One could argue therefore that this method would not be appropriate if you knew from other evidence or logical reasoning that there is a non-zero risk, i.e. that <span style="color: #000000">λ</span> >0.

> Macro (anchor)



The links to the Poisson Confidence Construction software specific models are provided here:

<details>
<summary>Crystal Ball</summary>

> Macro (inline-media-image)

 [Poisson_confidence_construction](/wiki/download/attachments/26579343/Poisson_confidence_construction-CB.xlsx?version=1&modificationDate=1506113500000&cacheVersion=1&api=v2)
</details>

<details>
<summary>@Risk</summary>

> Macro (inline-media-image)

 [Poisson_confidence_construction](/wiki/download/attachments/26581760/Poisson_confidence_construction-AtRISK.xlsx?version=1&modificationDate=1506097520000&cacheVersion=1&api=v2)
</details>

 

 

 

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