Consider two random variables X and Y with variances 4 and 9

Consider two random variables X and Y with variances 4 and 9 respectively.

a) What is the variance of (X–Y) when cor(X,Y)=0?

b) What is the variance of (X – Y) when cor(X, Y) = 0.5?

c) What is the variance of (X – Y) when cor(X, Y) = – 0.5?

Solution

As we know from properties of variance,

V(X-Y) = V(X) +V(Y) - 2Cov(X,Y)

a) V(X-Y) = 4+9-2(0) =13

b) V(X-Y) = 4+9-2(.5) =12

c) V(X-Y) = 4+9-2(-0.5) = 13+1 =14