"10 A natural ancillary statistic In most problems is the sample size . For example , let IN` he a random variable taking values 1 , 2 . ... with known probabilities p1 , p2 . .... where I'D; = 1 . Having observed N = ~, perform ~ Bernoulli trials with success probability* ` , getting & successes . ( a ) Prove that the pair ( * , N ) is minimal sufficient and N is ancillary for O. ( Note the similarity to some of the hierarchical models discussed in Section 1 . 4 . ) (b ) Prove that the estimator * / N is unbiased for A and has variance A ( 1 - 9 ) E ( 1 / N )