Question

3) Four statistically independent random variables, X, Y, Z, W have means of 2, -1, 1, -2 respectively, variances of X and Z are 9 and 25 respectively, mean-square values of Y and W are 5 and 20 respectively. Define random variable V as: V = 2X - Y + 3Z - 2W, find the mean-square value of V (with minimum math).

Answer 1

Mean Square Value of X = E(X²)

Given that

E(X) = Mean of X = 2

E(Y) = Mean of Y= -1

E(Z) = Mean of Z= 1

E(W) = Mean of W= -2

We have the formula for variance that

V(X) = E(X²) - [E(X)]² = 9

V(Z) = E(Z²) - [E(Z)]² = 25

Mean Square value of Y= E(Y²) = V(Y) + [E(Y)]² = 5, This imples V(Y) = 5 -1 = 4

Mean Square value of W= E(W²) = V(W) + [E(W)]² = 5, This imples V(W) = 20 - 4 = 16

Given that V = 2X - Y + 3Z - 2W and X, Y, Z and W are independent.

Hence Variance of V = 2² V(X) + V(Y) + 3² V(Z) + 2² V(W)

= 4 x 9 + 4 + 9 x 25 + 4 x 16 = 329

We have E(V) = 2 x E(X) - E(Y) + 3 x E(Z) - 2 E(W) = 2 x 2 + 2 + 3 x 1 - 2 x -2 = 13

Thus we have E(V²) = Mean square value of V = Variance of V + [E(V)]² = 329 + 169 = 498.