22. (i) Based on the conditional distribution of X 2 , . •. , X n given Xl = Xl in the model of Problem 20, there exists a UMP unbiased test of H : Po = PI against PI > Po for every a . (ii) For the same testing problem, without conditioning on Xl there exists a UMP unbiased test if the initial probability WI is assumed to be completely unknown instead of being given by the value stated in (i) of Problem 20. [The conditional distribution of X2 , • • • , X; given Xl is of the form C(XI ; Po, PI' s«. ql) p{IPtoqi'Q5°h(YI ' Y2' Z!. Z2) ' where YI is the number of times a 1 follows a 1, Yo the number of times a 1 follows a 0, and so on, in the sequence Xl' X2 , . .. , Xn • [See Billingsley (1961, p. 14).