Approximations and more PMFs and PDFs

1 Approximation of binomial with Poisson Consider the binomial distribution b(k, n, p) = n k p k (1 − p) n−k , 0 ≤ k ≤ n Assume that n is large, and p is small, but np → λ at the limit. For a fixed λ: b(0, n, p) = n 0 p 0 (1 − p) n = (1 − p) n = (1 − λ/n) n → e −λ b(1, n, p) = n 1 p(1 − p) n−1 = np 1 − p b(0, n, p) = λ 1 − λ/n b(0, n, p) → λe −λ b(2, n, p) = n 2 p 2 (1 − p) n−2 = n(n − 1)p 2 2(1 − p) 2 b(0, n, p) = n(n − 1)λ 2 2n 2 (1 − λ/n) 2 b(0, n, p) → λ 2 2 e −λ Continuing this way, we find that (when n is large) b(k, n, p) ≈ p(k, λ) = λ k e −λ k! where λ = np. While p(k, λ) represents an approximation for the binomial probability (b, n, p), p(k, λ) is a probability mass function on its own, known as the Poisson mass function.