Question

Suppose that X1, X2, . . . , Xn are independent identically distributed random
variables with variance σ2. Let Y1 = X2 +X3 , Y2 = X1 +X3 and
Y3 = X1 + X2. Find the following : (in terms of σ2)
(a) Var(Y1)
(b) cov(Y1 , Y2 )
(c) cov(X1 , Y1 )
(d) Var[(Y1 + Y2 + Y3)/2]

Answer 1

Since, X1, X2 & X3 are independent & identically distributed, so we have,

var(X1) = var(X2) = var(X3) = ².

And, cov(X1 , X2) = cov(X2 , X3) = cov(X1 , X3) = 0.

Now,

(a) var(Y1) = var(X2 + X3) = var(X2) + var(X3) + 2.cov(X2 , X3) = ² + ² + 0 = 2.²

(b) cov(Y1 , Y2) = cov(X2 + X3 , X1 + X3) = cov(X2 , X1) + cov(X2 , X3) + cov(X3 , X1) + var(X3) = 0 + 0 + 0 + ² = ².

(c) cov(X1 , Y1) = cov(X1 , X2 + X3) = cov(X1 , X2) + cov(X1 , X3) = 0 + 0 = 0

(d) var[(Y1 + Y2 + Y3)/2] = (1/4). [var(Y1) + var(Y2) + var(Y3) + 2.cov(Y1 , Y2) + 2.cov(Y1 , Y3) + 2.cov(Y2 , Y3)] = (1/4). (² + ² + ² + 0 + 0 + 0) = 3.²/4