3. (Bonus, 5pts) In this question, you will apply the EM algorithm to estimate the pa- rameters of the two-component exponential mixture model, where a random sample Y1, ..., Y, are i.i.d. with density fr(y) = phi(y) + (1 -p)fz(y), y>0. Let G; 6 {1, 2} be the unobserved membership of the ith observation. . pe (0, 1) is the mixing probability, i.e., the probability that G; = 1. . fily) = Me dis is the density of Y | G; = 1, i.e., the exponential distribution with rate > >0. . fz(y) = Age day is the density of Y | G; = 2, i.e., the exponential distribution with rate 12 > 0. An example is the whiteboard marker lifetime in Exam 1, where you generated random samples with p = .7, >1 = 2, 12 = .5. Here, we want to estimate the parameters (p, A1, and A2) using the EM algorithm, based on an observed sample y = y1, - - - ,ya- (a) Let g = (91, ---,9n) denote the unobserved group memberships of each observation. Provide an analytic expression of the complete data log-likelihood, ((p, A1, 12; y, g)- (b) Generate a random sample of size n = 1000 from the mixture model with p = .7, Al = 2, 12 = .5. Apply the EM algorithm to obtain marginal maximum likeli- hood estimates, p, d1, 12- (c) Under . p= .7, X1 = 2, 12 = .5, and . sample sizes n = 100, 1100, 2100, ..., 5100, obtain Monte Carlo estimates to the MSEs of p, Aj, A2. Plot the relationship be- tween n and the MSEs, and describe what you observe (i.e., do they increase, decrease, or remain unchanged, what they seem to approach)