Question

2. Suppose the variables Yi and Y have the following properties EQİ)-4, Var(h)-19, E(Y )-6.5, Var(Ya)-5.25, E(Y3%)-30 Calculate the following; please show the underlying work a) (3 pts) Cov(, ) b) (3 pts) Cov(41, 3%) c) (3 pts) Cov(41.5-½) (6 pts) Find the correlation coefficient between 1 + 3, and 3-2%

Answer 1

a) Cov(Y1,Y2) =E(Y1*Y2)-E(Y1)*E(Y2)=30-4*6.5=4

b)

as we know that Cov(aX+b,CY+d) =ac*Cov(X,Y)

hence Cov(4Y1,3Y2) =4*3*Cov(Y1,Y2) =12*4=48

c)Cov(4Y1,5-Y2)=4*(-1)*Cov(Y1,Y2)=-4*4=-16

d)

Cov(1+3Y1,3-2Y2) =3*(-2)*Cov(Y1,Y2) =-6*4=-24

Var(1+3Y1) =3²*Var(Y1)=9*19

Var(3-2Y2)=(-2)²*Var(Y2)=4*5.25

hence Correlation coefficient =Cov(1+3Y1,3-2Y2)/sqrt(Var(1+3Y1)*Var(3-2Y2))

=-24/sqrt(9*19*4*5.25)=-0.400501