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4. [13 marks] Suppose that X], .., X., is an independent random sample from a normal distribution with mean pi and variance o?. Also, suppose Y1, ..., Y,, is an independent random sample from a normal distribution with mean /2 and variance o?. Assume that the two samples are independent and that the N(u, a?) probability density function is exp _(I - 1 ) ? ] 202 (a) Write down the log-likelihood function for (/1, (62, 07) (b) What are the MLEs of #, and #2? (c) Using part (b), find the MLE of a" (d) Suppose we are interested in testing the hypothesis that the two means are both equal to zero: Ho : th - (2 -0 versus H1 : #1 * 12- Write down the number of parameters in the model under Ho and the number of parameters in the model under Hi- (e) For the hypotheses in part (d), write down the likelihood ratio test statistic and critical value for a test with significance level o - 0.05. [You do not need to give the numerical value of the critical value, just provide the appropriate distribution and percentile.]