4. Consider the expression n f (y|0) = [Ifi(yile), i=1 and define fi(yi|0) = f(yi|0, xi) where x; denotes a known "covariate" variable. In particular, let the xi's identify two groups, xi = 1 for i = 1, ..., k and xi = 0 for i = k + 1, ...,n. With 0 = (01, 02)', define di = 0102 ! "i so that di equals 01 if xi = 1) and equals 02 if xi = 0. Now consider f(yi|0, xi) = f(yildi) for each of the choices: (i) f(yilli) = die-digi, (ii) f(yilli) = Xhe di / yi!, (iii) f(yildi) = 2Niyie-day?. (a) Using Table 2.1, for each of the three choices identify the distribution with density f(yilli) when xi = 1 and also when xi = 0. (b) For each choice, simplify the product IIT_, fi(yi|0) so that as a function of 0 it is propor- tional to a function that depends only on some combination of Et-1g(yi), Et-k+19(yi), II_1 9(yi), and IIng(yi) where g(x) can be one of x, x2, or a!.