Example 19.LetX= (X, . . . , X) be conditionally IID Exp() given =.1nConsider computing the expected value ofXng(X) =.X+ +X1nTo do this, note thatg(X) is an ancillary statistic. Indeed, ifZ= (Z, . . . , Z) are1n1IID Exp(1) thenXd=Zand we see thatparenleftBigparenrightBig1XX1n1P(g(X)x) =P<+ ++ 1xXXnnparenleftBigparenrightBig1ZZ1n1=P<+ ++ 1xZZnnSince the distribution ofZdoes not depend onwe see thatg(X) is ancillary.The natural statisticT(X) =X+ +Xis complete (by the Theorem just1nproved) and minimal sufficient. By Basu's theorem (Theorem 13)T(X) andg(X)are independent. Hence,=E[X] =E[T(X)g(X)] =E[T(X)]E[g(X)] =nE[g(X)]n1and we see thatE[g(X)] =n.