A natural ancillary statistic in most problems is the sample size. For example, let N be a random variable taking values 1, 2,... with known probabilities p1, p2, . . ., where ∑pi = 1. Having observed N = n, perform n Bernoulli trials with success probability θ, getting X successes.
a. Prove that the pair (X, N) is minimal sufficient and N is ancillary for θ.
b. Prove that the estimator X/N is unbiased for θ and has variance θ(1 - θ)E(1/N).