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| 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.
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| 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 <math>\lambda</math>. 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.
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| Things I want to do:
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| *using the Poisson random number generator, generate <math>n</math>numbers using parameter <math>\lambda_0</math> and quantify
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| **how well a Poisson distribution with parameter <math>\lambda_0</math> fits the generated data and how this varies with <math>n</math>
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| **what parameter for the Poisson distribution <math>\lambda</math> best fits the generated data and find a standard deviation for how much the parameter varies
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| *repeat the above but with different <math>\lambda_0</math>. Try to find a relationship between <math>\lambda</math>, <math>\Delta\lambda</math>, and <math>n</math>
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| *for data generated with parameters <math>\lambda_0</math> and <math>n</math>, find how accurate are the Poisson distribution's predictions of what the count frequency for count number <math>k</math> is, and how this varies with <math>k</math>, <math>n</math>, and <math>\lambda</math>
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| For the test on how accurate the overall fit is, the dependent variable is <math>\Delta\lambda</math> and the independent variables are <math>n</math> and <math>\lambda</math>. For the test of the accuracy of the distribution's predictions about the frequency of a count number, the dependent variable is <math>\Delta P</math> and the independent variables are <math>k</math>, <math>n</math> and <math>\lambda</math>.
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| ==some planning==
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| The data sets can be very large if I become obsessive about it. I should establish a lower limit now.
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| How many times do I need to calculate <math>\lambda</math> before getting an accurate <math>\Delta\lambda</math> ?
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| Ack, I'll just use 50. That's enough for a variance, right?
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| ==Notebook Links== | | ==Notebook Links== |