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Download a pdf copy of this help file here |
|
Download a pdf copy of this help file here |
We assume that each observation is a Poisson random variable that has a
rate l. Letting X be the random variable
of the observed rate over the observation period (assuming time) t, it
will take a distribution given by:
(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, and unbiased, estimate for l. Switching around Equation 1, with X = a/t, we can get an uncertainty distribution for the true value of l:
(2)
This exactly equates to the non-parametric and parametric Bootstrap estimates of a Binomial probability. Equation 2 is awkward since it will only allow discrete values for l i.e. {0, 1/t, 2/t, ...}, whereas our uncertainty about l should really take into account all values greater than zero:
Figure 1: Example of Equation 2 estimate of l where a = 2, t = 4
It also makes no sense that l could be either zero (equivalent to the events cannot occur) when we have seen non-zero a, of course.