Physics307L F07:People/Mondragon/Poisson/Notebook

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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.

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.

Things I want to do:

  • using the Poisson random number generator, generate [math]n[/math]numbers using parameter [math]\lambda_0[/math] and quantify
    • how well a Poisson distribution with parameter [math]\lambda_0[/math] fits the generated data and how this varies with [math]n[/math]
    • 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
  • 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]
  • 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]

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].

some planning

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 [math]\lambda[/math] before getting an accurate [math]\Delta\lambda[/math] ?

Ack, I'll just use 50. That's enough for a variance, right?

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