Consider an event of probability p. The probability of n occurrences of this event out of N trials is given by the binomial distribution

Consider an event of probability p. The probability of n occurrences
of this event out of N trials is given by the binomial distribution

If p is small (p << 1), W_N (n) is very small, except for n << N. In this limit, show that we obtain the Poisson distribution,

where λ = np is the mean number of events. Check that P (n) is normalized. Calculate n and (∆n)² for this Poisson dis- tribution. Formulate a statistical problem to be solved in terms of this distribution.