4.49 Behboodian (1990) illustrates how to construct bivariate random variables that are uncorrelated but dependent. Suppose...

Transcribed Image Text:

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. 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.

Answer rating: 100% (QA)

Solution a Marginal Distributions The probability density function pdf of X Y is a weighted sum of two independent densities af1xg1y and 1af2xg2y To f... View the full answer