Theorem 1: If the probability p of success of a single trial approaches 0 while the number of trials n approaches infinity and the value μ = np stays fixed, then the binomial distribution B(n, p) approaches the Poisson distribution with mean μ.
Proof: Let be the pdf for a binomial distribution as described in the statement of the theorem. Then
Thus as p → 0 and n → , with μ = np fixed, we have
which completes the proof.