Question

19. In reviewing some data on smoking (X, number of packages of cigarettes smoked per year), income (Y, in thousands per year) and health (Z, number of visits to the family physician per year) for a sample of males, it is found that E(X) = 10, V(X) = 25, E(Y) = 50, V(Y) = 100, E(Z) = 6, V(Z) = 4, and Cov(X,Y) = -10, Cov(X,Z) = 2.5. Dr. N.A. Ively, a young statistician, attempts to describe the variable Z in terms of X and Y by the relation Z = X + cY, where c is a constant to be determined. Dr. Ively's methodology for determining c is to find the value of c for which Cov(X,Z) remains equal to 2.5 when Z is replaced by X + cY. What value of c does Dr. Ively find?

Answer 1

concx, z)=2.5 => eor ( x, X+ey) = 2:5 => cor(x,x) + e Cor(x,y) = 2:5 => vex)+ e cor(x, y) = 2.5 => 25+ eX(-10)=2.5 [cor(x,y)=-10 V(x)=257

C X (-10) - 2.5-25 2.5-as. 25 a - -10 = 2:25 c=2.25