Consider two independent random samples X1, . . . 1Xn and Y1, . . . , Km and assume they fol- low Binomial distributions: Xl; ~ Bino(n1, in) and Y; N Bin0(n2,1r2), where in and 11]} are unknown parameters of these Binomial distributions. Let 1(1rl, 71-2) denote the log-likelihood function from these two samples combined. (:1) Demonstrate that the log-likelihood for 111,711) from these two samples is given by 11. m roll, A2) = logbn) Z X+log(1rr1)2(n1Xi)+log{1r2)2 Yj+log(1rr2) 20123?)- 1:1 i=1 3:1 3:1 (b) Using the above log-likelihood to show that the maximum likelihood estimators (MLES) of m and Tr; are m $51 = 21:21 Xi; 7n: 25":1 Y} nnl mm (c) Derive the MLE for the ratio of an over 1T2, namely 9 = 1/113. (d) Next, you are required to perform a Wald test on the hypothesis Ho : in = 71'? and g = 9, where 7r? and N3 are two known values. (i) Find the expected information matrix for in and 7T2. (ii) Write down the asymptotic normal distributions for the MLEs if] and E2. (iii) Explain how to perform the Wald test for this hypothesis testing problem. 1{our answer should include: the test statistic formula and how to determine a critical value