Counting statistics of cosmic rays is explored using a LabVIEW based computer program on a computer connected to a National Instruments data acquisition unit. The program collects the interval times between successive coincident pulses from a stack of plastic scintillator detectors. Analysis of the data set allows students to see the distribution of N-pulse intervals for various values of N; this is the Erlang distribution. With N = 1, the Erlang
distribution is exponential with a characteristic time equal to mean count interval t. When N is increased, the distribution becomes peaked at Nt with a fractional width proportional to 1/?N. This is an instance of the central limit theorem. Students may also examine the data set according to the distribution of numbers of pulses recorded for a series of fixed-length intervals, which for random pulses follows the Poisson distribution. Again, as the length of the interval increases, the distribution conforms to the central limit theorem: it becomes normal with a well-defined mean and width, both related to mean and width of the underlying distribution.

The software also allows students to simulate pulse-intervals that follow a uniform distribution (e.g., any real number between 0 and 1 has equal probability) or a Gaussian one.  In these cases, one can see that the counts per fixed interval length do not follow the Poisson distribution, although all types become normal at longer intervals or greater numbers of pulses per interval length. This serves to drive home the point that cosmic ray counts are truly a Poisson process and also to illustrate the significant power of the central limit theorem-- that regardless of the underlying probability distribution function, when N becomes large the distribution becomes normal with a well-defined mean and width.

Presented at the 2013 AAPT Summer Meeting
in Portland, Oregon.
W36: Advanced Labs Workshop