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 $$\lambda$$. 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 $$n$$numbers using parameter $$\lambda_0$$ and quantify
 * how well a Poisson distribution with parameter $$\lambda_0$$ fits the generated data and how this varies with $$n$$
 * what parameter for the Poisson distribution $$\lambda$$ best fits the generated data and find a standard deviation for how much the parameter varies
 * repeat the above but with different $$\lambda_0$$. Try to find a relationship between $$\lambda$$, $$\Delta\lambda$$, and $$n$$
 * for data generated with parameters $$\lambda_0$$ 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 $$\lambda$$

For the test on how accurate the overall fit is, the dependent variable is $$\Delta\lambda$$ and the independent variables are $$n$$ and $$\lambda$$. For the test of the accuracy of the distribution's predictions about the frequency of a count number, the dependent variable is $$\Delta P$$ and the independent variables are $$k$$, $$n$$ and $$\lambda$$.

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 $$\lambda$$ before getting an accurate $$\Delta\lambda$$ ?

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

Notebook Links

 * Monday, December 17
 * Wednesday, December 26
 * Thursday, December 27