The purpose of this lab was to get an introduction to Poisson statistics, and to gain some familiarity with Matlab. Poisson statistics is a way of representing information using a probability measurement dependent only on the mean of a set of data. There are characteristic shapes for the Poisson distribution depending on where the mean is. If the mean is very close to zero, it is possible to see a curve with a large vale at x=0 that rapidly drops to zero as x increases. As the mean increases, a peak appears, but the amplitude of the peak decreases as the mean increases. Eventually, the Poisson curve looks like the Gaussian, and in fact the Poisson curve can be used to derive the Gaussian equation.

I used a radioactive Cobalt-60 source to generate a set of data centered around a mean number of counts recorded in a time interval, or a “bin”. Depending on how long I recorded the source, I could get a small mean or a much larger mean. The activity of the source was recorded by a photomultiplier tube sensor, which translated each event into a current. This current went to a pre-amplifier, which turned the current into a series of voltage pulses, which were further amplified by a delay line amplifier and displayed on an oscilloscope. A multi-channel analyzer (MCA) counted the number of pulses it recorded in a window of time. Thus each time the PMT recorded an event, it was translated into a voltage pulse and recorded by the MCA.

The MCA was a computer program, and after it had finished collecting a set of data I exported it into Matlab. Using this program, I was able to find the mean number of events per time interval, and to graph this data. I varied the time intervals, and the resultant graphs are shown below.

One of the main points of this experiment was to compare the Poisson distribution to other distributions. I compared the Poisson distribution to the Gaussian and to the Binomial distributions.

It only makes sense to compare the Gaussian fit to the Poisson fit when the mean is fairly large, since the Poisson fit at low means is very different from the fit at large means. I compared the fits at a mean of 20777.7, when the bin size was 10 s. While there are slight differences in the fits, they both closely adhere to the data.

I also compared the Poisson distribution to the Binomial distribution. The Binomial distribution, unlike either the Poisson distribution or the Gaussian, deals with data that has either two responses: 0 or 1. It was necessary, therefore, for the bins to either record no events or one event. This was accomplished by decreasing the bin size to 5 us, so the mean was 0.0097. Such a distribution is shown in Figure 5. Notice that most of the bins recorded zero events, with a small number recording only one event. No bins recorded more than one event.

We then examined how many events we were likely to get with a bin size of 5 us. This was accomplished by obtaining 300 sets of data (6 sets were taken at a time) and plotting the number of events in each 10000-bin set. This was then compared to the binomial distribution. The Binomial Distribution is simply another way to analyze this data.

This lab was interesting because it explored a number of ways to represent the data associated with random events. While some of these fits, like the Gaussian, I had worked with before, there were new fits such as the Binomial distribution that I had never encountered before. This lab also taught me a lot about fitting these distributions in Matlab. It was an interesting way to end the semester!