imal sufficient statistic. whether the set ()1, . . . , Yn-1) is independent of the min- 412 A natural ancillary statistic in most problems is the

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imal sufficient statistic.whether the set ()1, . . . , Yn-1) is independent of the min-412 A natural ancillary statistic in most problems is the sample size. For example, let Non. Is [X | a sufficient statistic?be a random variable taking values 1, 2, . .. with known probabilities p1, p2, . .., wherewith densitiesEpi = 1. Having observed N = n, perform n Bernoulli trials with success probability0, getting X successes.,-2 10 W 100 > > 00-,630)< 10. anoilsanglapand to jog al(a) Prove that the pair (X, N) is minimal sufficient and N is ancillary for 0. (Notethe similarity to some of the hierarchical models discussed in Section 4.4.)for 0.(b) Prove that the estimator X/N is unbiased for 0 and has variance 0(1 -0) E(1/N).$13 Suppose X1 and X, are iid obse