# Physics307L F07:People/Mondragon/Poisson/Notebook

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]\displaystyle{ \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]\displaystyle{ n }[/math]numbers using parameter [math]\displaystyle{ \lambda_0 }[/math] and quantify
- how well a Poisson distribution with parameter [math]\displaystyle{ \lambda_0 }[/math] fits the generated data and how this varies with [math]\displaystyle{ n }[/math]
- what parameter for the Poisson distribution [math]\displaystyle{ \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]\displaystyle{ \lambda_0 }[/math]. Try to find a relationship between [math]\displaystyle{ \lambda }[/math], [math]\displaystyle{ \Delta\lambda }[/math], and [math]\displaystyle{ n }[/math]
- for data generated with parameters [math]\displaystyle{ \lambda_0 }[/math] and [math]\displaystyle{ n }[/math], find how accurate are the Poisson distribution's predictions of what the count frequency for count number [math]\displaystyle{ k }[/math] is, and how this varies with [math]\displaystyle{ k }[/math], [math]\displaystyle{ n }[/math], and [math]\displaystyle{ \lambda }[/math]

For the test on how accurate the overall fit is, the dependent variable is [math]\displaystyle{ \Delta\lambda }[/math] and the independent variables are [math]\displaystyle{ n }[/math] and [math]\displaystyle{ \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]\displaystyle{ \Delta P }[/math] and the independent variables are [math]\displaystyle{ k }[/math], [math]\displaystyle{ n }[/math] and [math]\displaystyle{ \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]\displaystyle{ \lambda }[/math] before getting an accurate [math]\displaystyle{ \Delta\lambda }[/math] ?

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