Question 3 [16 marks] Two independent random samples X1, ..., X, and Y1,...; Ym follow Poisson distributions: Xi ~ Poi()1) and Yi ~ Poi(12), where > > 0 and > > 0 are two unknown parameters. Let 1(1, 12) denote the log-likelihood function from these two samples. (a) Show that the log-likelihood function for A, and 12 is given by I(Al, A2) = -ndi - maz + ()Xi)log di + () Y;) log A2. 1= 1 j=1 (b) Show that the maximum likelihood estimators (MLEs) of A, and 12 are m (c) Hence find the MLE of 8 = M/ 12. (d) Find the expected information matrix for >, and 12. (e) Write down the asymptotic normal distributions for the MLEs A, and Az. (f) Compute the maximum likelihood estimate of A where now Al = 12 = 1. (g) Explain how to perform the log-likelihood ratio test for testing the hypothesis: Ho : A1 = 12 V.S. H1 : not Ho