# Poisson Distribution

Steve Koch 01:15, 22 December 2010 (EST):Good notebook

### Materials

• Spectech Universal Computer Spectrometer power supply, UCS 30
• Spectrum Techniques UCS30 Software

### Procedure

There is no setup to this laboratory. Turn on the Spectech, initiate the Spectrum Techniques software. The software is not intuitive though and this requires some explanation.

• Under MODE select PHA(AmpIn)
• Under SETTINGS select Amp/HV/ADC. These settings include a High Voltage, Voltage Polarity, Amp in Polarity, Course Gain, Fine Gain, Conversion Gain, LLD, ULD, and Peak Time.
• Once the settings are taken care of select MODE again and select MCS(internal) then hit the green GO button and watch the blue screen get spattered with white dots. Once data the program runs its course, the data can be saved and later analysed.

### Settings

Settings

• High Voltage: On at 1200V
• Voltage Polarity: Positive
• AMP In Polarity: Positive
• Coarse Gain: 12
• Conversion Gain: 2048

### Theory

A Poisson process is a process that describes many natural events ranging from rainfall to radioactive decay. 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
$\Pr = \frac{e^{-\lambda t} (\lambda t)^k}{k!}$
The probability on an event happening in one measured period is
$\Pr = \frac{e^{-\lambda} (\lambda)^k}{k!}$
Here λ is the expected value (the average value) and k 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.

### Data Collection

Background radiation data was collected in 10ms,20ms,40ms,80ms,100ms,200ms,400ms,800ms,and 1s time intervals. Each interval was then sub-sectioned into 2049 sub-intervals. In these sub-intervals the number of radiation events was observed and recorded.

The above probability data is this function applied to the collected data.

    function [P,x1,y,sy,sP,error] = Poisson(X)
x=X(:,3);
x1=x;
mu=mean(x);
P=exp(-mu)*(mu.^x1)./(factorial(x1));
for i=1:max(x1)+1;
ind1=find(x1==i-1);
x1(ind1(1:end-1))=[];
P(ind1(1:end-1))=[];
end
[x1,IX]=sort(x1);
P=P(IX);
length(x1);
y=zeros(1,length(x1));
for j=1:length(x1);
y(j)=length(find(x1(j)==x));
end
y=y./sum(y);
sy=std(y./sum(y));
sP=std(P);
error=100*abs((sy-sP)/sy);
end


As one can see the above probability data is overly repedative ie there are many data point that are repeated. Therefore, it is helpful to see a visual representation of the data to better ascertain if it represents a Poisson event. Plotted here are the intervals plotted so that the probability vs number of events is easy to see. One will notice that at low time intervals the probability for an event happening looks like a simple exponential graph. This is because the probability of an event occuring averages out to be zero. As the intervals lengthen in time one observes a shift in the maximum probability and the graphs go from looking like simple exponentials to Gaussian like curves. Indeed at the longest time intervals the Gaussian nature of the curves is obvious. This indicates that although the event is random in nature there is an overall temporal patter of events per unit time that can be calculated. This is the definition of a Poisson process.