Question

(3) Unbiased Estimator Y is distributed N( 14, a2 ). Weighted sample mean is defined in the following: N Σ4r Y = - S.Σα =Ν (a) Please prove weighted sample mean is unbiased

Answer 1

If sample mean is unbiased the expected value of sample mean should be equal to the actual mean of the distribution.

It is given that the distribution follows Normal distribution with mean  $\mu$

Let us calculate the expected value of the weighted sample mean and check if it is equal to $\mu$

a,Y Y N

ΣΥ Ε(Υ )-Ε(Σ E( Ν

ΣαΕΥ) Ε(Υ) . Ν

Σαμ Ε(Υ) . Ν.

.............. (1)

ai E(Y) N

It is given that

aiN

Therefore from (1)

E(Y) N

Therefore, sample mean given in the question is unbiased.