Physics307L:People/Wilkinson/Poisson

Poisson Distribution Summary

Steve Koch 01:16, 22 December 2010 (EST):Good job on this lab and good job turning it into formal report.

Motivation

The goal of the scientist is to describe and quantify natural phenomena. In an ideal world, events and processes would be easy to describe and outcomes would be exactly predictable. Unfortunately, no such world exists, most of the time scientists can only assign probabilities to outcomes and maybe describe the possible error in their predictions. The need for a good probabilistic model is the basis for probability distributions. The Poisson distribution is a probability distribution that can be used to model many natural processes. The most apparent example is that of rainfall. If one were to consider the average number of raindrops that hit the ground in a given time he would expect it to be proportional to the time considered (the longer the time interval the more raindrops will hit the ground). Though at very short intervals he would also expect very few drops to hit the ground. As the interval is lengthened the average number of drops that hit the ground should increase and he error away from this average should also increase. With this mental picture in mind we will now consider radiation events in the lab and try to see if they fit this probability model.

Theory

To be considered a Poisson process, the process in question must be composed of continuous events that are independent of one another. The events can be observed at different time intervals and is described by an exponential distribution. The statistical validity comes in when the random events have some average rate of occurring over a significantly large time interval ie at small time intervals the events seem to be randomly distributed. As larger and larger intervals are considered one can calculate an average number of events and the distribution tends to look Gaussian.
The probability of an event happening is defined by
[math]\displaystyle{ \Pr = \frac{e^{-\lambda t} (\lambda t)^k}{k!} }[/math]
The probability on an event happening in one measured period is
[math]\displaystyle{ \Pr = \frac{e^{-\lambda} (\lambda)^k}{k!} }[/math]
Here [math]\displaystyle{ \lambda }[/math] is the expected value (the average value) and [math]\displaystyle{ k }[/math] is the number of observed events.
To determine if a process is Poisson one must simply take measurments of the events in the process at differing time periods and then compute and plot the probabilities.

Results

First we collected data as outlined in the Lab Notebook. The raw data was fit to a Poisson distribution and the standard deviations of both were taken.

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Here it is shown that the percent error between the Poisson fit and the normalized raw data is very low. Therefore, this process must be Poisson in nature.

Acknowledgments

I got the idea for calculating the percent error between the standard deviations of the normalized raw data and the Poisson fit data from Brian Josey.