Particle Decay is a Random Phenomena: Part III

Now that we understand what it means for a process to be random let's delve into the mathematics. The most important distribution to understand is the Poisson distribution. The probability of exactly n Poisson occurrences during a time interval where were expected is:

To satisfy the Poisson conditions events must be occurring with an average rate and be independent of each other, decaying pions certainly qualify. There are some interesting and useful features of this distribution. For instance the mean ()and the variance () are equal to . That means the standard deviation of a Poisson distributed data set is .

Say we measure 1,000,000 pions decays. We now want to state our measurement of the branching ratio (). How do we determine the statistical error of our measured value ()? Since we know the decays obey Poisson statistics we simply use the features we learned above combined with standard statistical estimation rules:

http://en.wikipedia.org/wiki/Standard_error_(statistics)

The important thing to note is that the "one over square root of N" has nothing to do with Poisson statistics but rather is a feature of sampling error.