2. Two independent random samples X1, . . . , Xn and Y1, . . . , Ym follow Poisson distributions: Xi P oi() and Yi P oi(), where 0 and 0 are two unknown parameters. Suppose we wish to test the hypothesis H0 : = 0, = 0 v.s. H1 : not H0, where 0 and 9 are given values. Let l(, ) denote the log-likelihood function from these two samples.

(a) Show that the log-likelihood function is given by l(, ) = n m + log Xn i=1 Xi + Xm j=1 Yj ! + log Xm j=1 Yj . (b) Hence show that the maximum likelihood estimators of and are b = Pn i=1 Xi n ; b = Pm j=1 Yj/m Pn i=1 Xi/n . (c) Derive the likelihood ratio test statistic for testing this hypothesis. (d) Specify the rejection region given by the likelihood ratio test.