Physics307L:People/Mondragon/Poisson

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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 poissrnd() function included in Gnu Octave vers. 3.0.0 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 ${\displaystyle \lambda }$.

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