48 Derivation from Binomial Distribution Wikipedia: This limit is sometimes known as the law of rare events, since each of the individual Bernouilli event rarely triggers. The name may be misleading because the total count of success events in a Poisson process need not be rare if the parameter λ is not small. For example, the number of telephone calls to a busy switchboard in one hour follows a Poisson distribution with the events appearing frequent to the operator, but they are rare from the point of the average member of the population who is very unlikely to make a call to that switchboard in that hour. The proof may proceed as follows. First, recall from from calculus that:and the definition of the Binomial distribution:If the binomial probability can be defined such that p = λ / n, we can evaluate the limit of P as n goes large:The F term can be written as:and then note that, since k is fixed, this is a rational function of n with limit 1.