User:Kirstin Grace Harriger/Notebook/Physics 307L/Poisson Statistics

Steve Koch 03:48, 21 December 2010 (EST):Very good notebook and nice data.

Lab 04: Poisson Statistics

Concepts

• Poisson Distribution: The Poisson Distribution is a discrete probability distribution. It expresses the probability of a number of discrete events occurring within a time frame if the timing of each event is independent of the timing of the previous event and there is a know average rate of events. It is assumed that no two events are simultaneous. If the expected number of occurrences in this interval is [math]\displaystyle{ \lambda }[/math] then the probability that there are exactly k occurrences is equal to

[math]\displaystyle{ f(k; \lambda)=\frac{\lambda^k e^{-\lambda}}{k!} }[/math]

In the Poisson Distribution, [math]\displaystyle{ \lambda }[/math] is the mean number of occurrences, and [math]\displaystyle{ \sqrt{\lambda} }[/math] is the standard deviation.

Safety Concerns

• Be aware of shock hazards

Equipment

• UCS 30 software on the computer
• combined PMT scintillator inside 4 short walls of lead bricks
• Spectech Universal Computer Spectrometer power supply

Procedure

Set Up

1. Turn on the computer and log-in if necessary,
2. Turn on the Spechtech, it has to be turned on first before the software,
3. Double click on the icon for the software on the desktop of the computer,

Data Collection

1. Under mode select "PHA (Amp In)"
2. Under Settings Select "High Voltage On", and set it to an appropriate value, we used 1200 V. This value is used to adjust the sensitivity of the detector, and a higher voltage will decrease the sensitivity to only the most energetic radiation
3. Under mode, select "MCS (Internal)"
4. Under Settings, select MCS, and then pick your appropriate dwell time, which is how large each bin is for the number of events counted.
5. To collect data, hit the green "Go button" and let it run its course
6. When it stops, save it to a file or USB drive, but save it as a "comma separated variable (*.csv)"
7. Import it into Google docs

The values for our dwells times were: 10, 20, 40, 80, 100, 200, 400, 800 ms and 1 s.

This is the Spectech Universal Computer Spectrometer power supply

This is the combined PMT scintillator inside 4 short walls of lead bricks

Data

Sheet 1 in each spreadsheet contains the data from the program and Sheet 2 contains the following: Min and Max count of events for a dwell time, the average of the counts per dwell time, the theoretical (root of the average) and the calculated standard deviation for the average as well as a percent error between the two, the prediction for a one second dwell time, and the number of dwell times (windows) with a given number of events.

{{#widget:Google Spreadsheet |key=0AoEBBJie7L2cdGw2Nnhub3FOYUFhZmZYQ2xna253WXc |width=375 |height=350 }}{{#widget:Google Spreadsheet |key=0AoEBBJie7L2cdGxLNG5aLU1IRUU0Tk96TkZCTFVnbFE |width=375 |height=350 }} {{#widget:Google Spreadsheet |key=0AoEBBJie7L2cdGtlb3NrdXdlS19zZ3hkZC1zaUw1VEE |width=375 |height=350 }} {{#widget:Google Spreadsheet |key=0AoEBBJie7L2cdGF1amVkOS0zZVhfS1I5S3p5RlZNdkE |width=375 |height=350 }} {{#widget:Google Spreadsheet |key=0AoEBBJie7L2cdEpvQnh5aWM5amtOVFhNam1CQ0RNRnc |width=375 |height=350 }} {{#widget:Google Spreadsheet |key=0AoEBBJie7L2cdEE0RkpMN0pSeDlHT0E3V21aTEFsWlE |width=375 |height=350 }} {{#widget:Google Spreadsheet |key=0AoEBBJie7L2cdGlpYnBXcFRQdFY1OVBSMnJXRFkyREE |width=375 |height=350 }} {{#widget:Google Spreadsheet |key=0AoEBBJie7L2cdEJsWFRFLV91WFFaajhOUzliUGNMQ3c |width=375 |height=350 }} {{#widget:Google Spreadsheet |key=0AoEBBJie7L2cdFU5NmdVUFZJYWdTbFo1ejJMOWhOVHc |width=375 |height=350 }}

Analysis

Each graph shows a given number of events for a dwell time on the x-axis, and the number of time periods in which a given number of events was recorded on the y-axis. If there were 4 dwell times in which 3 events occurred, then 3 is the x-coordinate and 4 is the y-coordinate. Each graph shows the results from a different dwell time. The dwell times from left to right are 10, 20, 40, 80, 100, 200, 400, 800 ms and 1 s. As the dwell time increases, the curve becomes more symmetric and the peak moves to the right. This is unique of the Poisson Distribution, and a good qualitative test.

To be sure our data represents a Poisson Distribution we compared the standard deviation calculated directly from the data to the root of the mean of the data. The root of the mean of the data is the standard deviation for the Poisson Distribution by definition, so the two numbers should be very close to each other. This is a good quantitative test.

{{#widget:Google Spreadsheet |key=0AoEBBJie7L2cdDJtZUF5SnctekZYWFFyalFoZFdMdWc |width=500 |height=300 }}

The percent differences are all very small, with the largest being 5.13% and the smallest being 0.18%. This shows that our data does represent a Poisson Distribution.

Resources

1. Prof. Gold's Lab Manual

1. The Wikipedia article on the Poisson Distribution

Collaboration

1. My lab partner Brian Josey