The desire to perform these experiments arose from arguments about how accurately a Poisson probability distribution could fit data that should most probably follow the Poisson distribution. One thing I see immediately is that the Poisson distribution can not fit data collected from a finite number of counting experiments exactly but as the number of counting experiments preformed approaches infinity, the data should fit the distribution more and more tightly.
I will be using the poisson_rnd() function included in Gnu Octave vers. 2.1.73 to generate random numbers with a Poisson distribution with parameter λ. Gnu Octave is open source, so it shouldn't be too difficult to find the source code and examine how these numbers are generated for those who are curious. I am using this number generator to model counting experiments.
Things I want to do:
- using the Poisson random number generator, generate nnumbers using parameter λ0 and quantify
- how well a Poisson distribution with parameter λ0 fits the generated data and how this varies with n
- what parameter for the Poisson distribution λ best fits the generated data and find a standard deviation for how much the parameter varies
- repeat the above but with different λ0. Try to find a relationship between λ, Δλ, and n
- for data generated with parameters λ0 and n, find how accurate are the Poisson distribution's predictions of what the count frequency for count number k is, and how this varies with k, n, and λ
For the test on how accurate the overall fit is, the dependent variable is Δλ and the independent variables are n and λ. For the test of the accuracy of the distribution's predictions about the frequency of a count number, the dependent variable is ΔP and the independent variables are k, n and λ.
The data sets can be very large if I become obsessive about it. I should establish a lower limit now.
How many times do I need to calculate λ before getting an accurate Δλ ?
Ack, I'll just use 50. That's enough for a variance, right?