# Cumulative confidence construction estimate for the Poisson intensity

The confidence we have that the true value of the intensity is less than or equal to λ is:

P(X>α; λ) + ½ P(X=α; λ)

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

=1-POISSON(α,λ,1)+0.5*POISSON(α,λ,0)

We can use this to construct a cumulative confidence distribution by testing a range of values of λ: Figure 1: Cumulative distributions of estimate of λ for varying number of observations α

The points used to construct the above plot can be fed into a Custom Distribution to construct a Cumulative Distribution 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 α = 0 starts at 0.5. That means the distribution is assigning 50% confidence to λ = 0, and the remaining 50% confidence to all other values of λ. 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 λ=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 λ >0.

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

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