1.18 For a sequence of independent Bernoulli trials, let Y be the number of successes before the kth failure. Explain why its probability mass function is the negative binomial, $$p(y) = \frac{(y + k - 1)!}{y!(k - 1)!} \pi^y (1 - \pi)^k$$, y = 0, 1, 2, ....
[For it, E(Y) = kπ/(1 – π) and var(Y) = kπ/(1 - π)², so var(Y) > E(Y); the Poisson is the limit as k → ∞ and π→ 0 with kπ = μ fixed.]