ABSTRACT
Exercise 4.1. To assess whether there is genetic predisposition to becoming a cigarette smoker, epidemiologic studies have been done in which sets of monozygotic twins separated at birth and raised to adulthood in totally different environments are located and personally interviewed regarding their current smoking habits. For the i-th adult member of such a set of monozygotic twins (i = 1, 2), let Xi = 1 if that adult member is currently a smoker, and let Xi = 0 if not. Because of the distinct possibility that responses from monozygotic twins may tend to be correlated, a certain biostatistician is not willing to assume that X1 and X2 are independent random variables. So, she suggests using the following two-parameter bivariate discrete probability distribution pX1,X2(x1, x2) for X1 and X2, where θ > 0 and where 0 < π < 1:
pr[X1 = X2 = 0] = K(1− π)2; pr[(X1 = 1) ∩ (X2 = 0)] = pr[(X1 = 0) ∩ (X2 = 1)] = Kπ(1− π)θ;
pr[X1 = X2 = 1] = Kπ 2.
