Calculate the variance of a point estimator Let X1 and X2 be

Calculate the variance of a point estimator

Let X1 and X2 be two independent random observations where E(X1) = mu and E(X2) = 2mu, but V(X1) = V(X2)= 8. Let hat mu1 be a point estimator of mu where hat mu1 = X1 + X2/2. 3. (5 points) Calculate the variance of hat mu1.

Solution

u1_hat = (X1 + X2) / 2
variance of the point estimator:
V(u1_hat) = V((X1 + X2) / 2)
= ((1/2)^2) V(X1 + X2)
= (1/4) (V(X1) + V(X2))
= (1/4) * (8 + 8)
= 4