Question

8)

X1X1 and X2X2 are two random variables that are jointly distributed such that E(X1)=8E(X1)=8, E(X2)=12E(X2)=12, Var(X1)=2Var(X1)=2, Var(X2)=3Var(X2)=3 and Cov(X1,X2)=−1Cov(X1,X2)=−1. Let Z=5X1−4X2−5Z=5X1−4X2−5.

1. Compute E(Z)E(Z).

2. Compute Var(Z)Var(Z).

3. Compute Corr(X1,X2)Corr(X1,X2).

Show all your working.

Answer 1

Here, X1 & X2 are two random variables that are jointly distributed such that E(X1) = 8,Var(X1) = 2,E(X2) = 12,Var(X2) =3 & Cov(X1 , X2) = -1.

Here, Z = 5X1 - 4X2 -5

So, E(Z) = E( 5X1 - 4X2 -5 )

= 5E(X1) - 4E(X2) - 5 ....................... (By the property of expectation, E(aX + b) = aE(X) + b)

= (58) - (412) - 5

= 40 - 48 - 5

E(Z) = -13

V(Z) = V( 5X1 - 4X2 -5 )

= 25V(X1) - 16V(X2) ............................(By the property of variance, V(aX + b) = a2V(X) )

= (252) - (163)

= 50 - 48

V(Z) = 2