# Poisson Statistics

## Purpose

To measure the random cosmic radiation and see if that random collection yields the Poisson distribution, and to learn about the Poisson distribution.

## Materials

• A Thallium doped sodium iodine crystal scintillator
• A Photomultiplier tube
• A preamp, amplifier and discriminator ("PAD")
• A PC data acquisition card with "hydra breakout cable"
• Multichannel Analyzer Software
• A NIM-bin
• A High-Voltage DC power supply for the PMT
• Excel

## Procedure

The sodium detector measures and sends a signal to the computer every time cosmic radiation hits it. The MCA computer device in the multichannel scaling mode counts the number of incident pulse signals in a certain dwell time and then jumps to the next bin and does the same. In the MCA program you can control the number of bins and the length of the dwell time.

## Poisson distribution

• The poisson distribution is a event probability distribution where the average time per event is known but the occurence of an event is independent of the time of last event's occurance. (see here for more)
• The poisson distribution is given by:

$f(k;\lambda)=\frac{\lambda^k e^{-\lambda}}{k!},\,\!$

where

• e is the e (mathematical constant) base of the natural logarithm (e = 2.71828...)
• k is the number of occurrences of an event - the probability of which is given by the function
• λ is a positive real number, equal to the expected number of occurrences that occur during the given interval.

## Gaussian or Normal Distribution

• The gaussian probablity function is given by:

$\frac1{\sigma\sqrt{2\pi}}\; \exp\left(-\frac{\left(x-\mu\right)^2}{2\sigma^2} \right) \!$

## Results

• The following images are histograms of the frequency versus the number of incident pulse readings that the MCA program counted. Each graph then has the same points plotted out in a gaussian distribution and a poisson distribution for the mean and the standered deviation of the data.
• The smaller dwell time data more accurately matches the Poisson distribution, as the dwell time increases the gaussian fit becomes more and more reasonable, the Poisson distribution is always a closer fit in the data that I gathered from purely a visual perspective.