4.49 Behboodian (1990) illustrates how to construct bivariate random variables that are uncorrelated but dependent. Suppose that f1, f2, 91, 92 are univariate densities with means , 2, 1, 2, respectively, and the bivariate random variable (X, Y) has density (X,Y) ~ afi(x)g(y) + (1 a) f2(x)g2 (y), where 0 < a < 1 is known. (a) Show that the marginal distributions are given by x(x) = a(x) + (1 a) 2(x) and fy(x) = ag (y) + (1 a)92(y). - (b) Show that X and Y are independent if and only if [1 (x) 2(x)][91(y) 92(y)] = 0. (c) Show that Cov(X, Y) = a(1a)[1 2][12], and thus explain how to construct dependent uncorrelated random variables. (d) Letting f1, f2, 91, 92 be binomial pmfs, give examples of combinations of parame- ters that lead to independent (X, Y) pairs, correlated (X, Y) pairs, and uncorre- lated but dependent (X, Y) pairs.