# 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 $\displaystyle{ \lambda }$. 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 $\displaystyle{ n }$numbers using parameter $\displaystyle{ \lambda_0 }$ and quantify
• how well a Poisson distribution with parameter $\displaystyle{ \lambda_0 }$ fits the generated data and how this varies with $\displaystyle{ n }$
• what parameter for the Poisson distribution $\displaystyle{ \lambda }$ best fits the generated data and find a standard deviation for how much the parameter varies
• repeat the above but with different $\displaystyle{ \lambda_0 }$. Try to find a relationship between $\displaystyle{ \lambda }$, $\displaystyle{ \Delta\lambda }$, and $\displaystyle{ n }$
• for data generated with parameters $\displaystyle{ \lambda_0 }$ and $\displaystyle{ n }$, find how accurate are the Poisson distribution's predictions of what the count frequency for count number $\displaystyle{ k }$ is, and how this varies with $\displaystyle{ k }$, $\displaystyle{ n }$, and $\displaystyle{ \lambda }$

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

## 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 $\displaystyle{ \lambda }$ before getting an accurate $\displaystyle{ \Delta\lambda }$ ?

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