6.34 Consider the model in Exercise 6.12. Show that the Formal Likelihood Principle implies that any conclusions about # should not depend on the fact that the sample size n was chosen randomly. That is, the likelihood for (n,x), a sample point from Exercise 6.12, is proportional to the likelihood for the sample point x, a sample point from a fixed-sample-size binomial(n,1) experiment. 6.12 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 0, getting X successes. (a) Prove that the pair (X, N) is minimal sufficient and N is ancillary for 0. (Note the similarity to some of the hierarchical models discussed in Section 4.4.) (b) Prove that the estimator X/N is unbiased for 0 and has variance 0(1 - 1)E(1/N)