\fMixture model: The le g1ucose.dat contains the plasma glucose con- centration of 532 females from a study on diabetes (see Exercise 7.6). a) b) Make a histogram or kernel density estimate of the data. Describe how this empirical distribution deviates from the shape of a normal distribution. Consider the following mixture model for these data: For each study participant there is an unobserved group membership variable X,- which is equal to 1 or 2 with probability p and 1 p. If X,- = 1 then Y, N normal(91, of), and if X,- = 2 then Y, N normal(92, 0%). Let p N beta(a,b), 6,- ~ normal(,ug,*rg) and 1/03- ~ gamma(y0/2,ygo/2) for both 3' = 1 and j = 2. Obtain the full conditional distributions of (X1, . . . ,X,,), p, 61, 62, of and 0%. Setting a = b = 1, Mo = 120, T02 = 200, org = 1000 and yo 10, implement the Gibbs sampler for at least 10,000 iterations. Let 98 = min{9s),9g3)} and 98; = max{9s),93)}. Compute and plot the autocorrelation functions of 98 and 68, as well as their effective sample sizes. For each iteration s of the Gibbs sampler, sample a value m N binary(p(3)), then sample 17(3) w normal(933),cr(s)). Plot a his- togram or kernel density estimate for the empirical distribution of 17(1),. . . ,17(S), and compare to the distribution in part a). Discuss the adequacy of this two-component mixture model for the glucose data