User:Garrett E. McMath/Notebook/Junior Lab/2008/11/24
|Poisson Statistics|| Main project page|
SJK Incomplete Feedback Notice|
SJK 17:30, 17 December 2008 (EST)
Data by Paul Klimov and Garrett McMath
Poisson statistics are crucial in our understanding of many phenomena. The famous distribution gives us the probability that a number of events will happen given that the event of interest is happening at a fixed average rate. The Poisson distribution is derived directly from the binomial distribution. The binomial distribution tells us the probability of k successes after n trials, each of which occurs with a probability p. The distribution is given as follows :
The expected value of successes is given by the product of the number of trials and the probability of success. I will denote this quantity by the parameter lambda, which will become convenient, as we will see later:
Using the new parameter, the binomial distribution can be re-written as follows:
In the limit of small probability and many trials (low p and large n), the above distribution approaches the following distribution:
This is the Poisson distribution. To learn more about the distribution, we must discuss several important statistical quantities. The mean can be computed as follows, returning:
The variance is then computed as follows (the sums sum over all values of k starting with 0):
Clearly the poisson distribution has a very interesting feature, that its mean and its variance are the same value -- the expectation parameter lambda. I will be exploiting this fact throughout this lab, and my data analysis will be based directly on this feature. Another important mention is that the Poisson distribution merges with the normal distribution (or gaussian) for large expectation values (lambda large). The famous gaussian distribution is given by the following formula,where sigma is the standard deviation and mu is the mean. :
In this lab we will be will be studying the rate of cosmic ray bombardment, a phenomenon which should be described by the Poisson distribution. The rays will be detected with a NaI scintillation counter. As rays strike the detector, they ionize a compound which becomes fluorescent upon ionization. The fluorescence is then picked up with a photomultiplier tube (PMT), which amplifies the signal, and gives us a reading. The reading will be interpreted by a computer operator.
Everything was wired according to Professor Gold's manual. Although the wiring looked a bit complicated, it amounted to us hooking up the NaI counter to a computer, where our data could be interpreted by a computer operator. All settings from there were adjusted on the computer operator. The software, although a bit outdated, actually worked quite well after we figured out how to use it.
The first important setting on the operator is the channel setting. Each channel (also bin) acts as a separate detector, in a sense. After the operator starts acquiring data from the detector, it starts counting the number of events into the first bin only. After a certain dwell time has elapsed, the operator switches to the next bin, and starts counting for the same length of time. The process continues until the operator has reached the last bin, at which point it can either repeat the process or stop taking data. For each trial, we had the operator set to use 256 channels. Dwell times were varied, as were the number of passes.
Due to the huge data outputs, I will have to link to the data. Although all original files were in ASCII format, everything was converted to excel. During the first few trials we were getting used to the software and so we did not take down the dwell times, or number of passes, unfortunately. Because we have a lack of information about those trials, I will not use them for any large conclusions.
Possible Sources of Error
Post Experimental AnalysisSJK 17:30, 17 December 2008 (EST)