9.3Distribution of the Estimators 357
=
α
∑
i
(xi−x)+β
∑
i
xi(xi−x)
∑
i
xi^2 −nx^2
=β
[∑
i
xi^2 −x
∑
i
xi
]
∑
i
x^2 i−nx^2
since
∑
i
(xi−x)= 0
=β
ThusE[B]=βand soBis an unbiased estimator ofβ. We will now compute the variance
ofB.
Var(B)=
Var
(n
∑
i= 1
(xi−x)Yi
)
(n
∑
i= 1
xi^2 −nx^2
) 2
=
∑n
i= 1
(xi−x)^2 Var(Yi)
(n
∑
i= 1
xi^2 −nx^2
) 2 by independence
=
σ^2
∑n
i= 1
(xi−x)^2
(n
∑
i= 1
xi^2 −nx^2
) 2
=
σ^2
∑n
i= 1
xi^2 −nx^2
(9.3.2)
where the final equality results from the use of the identity
∑n
i= 1
(xi−x)^2 =
∑n
i= 1
xi^2 −nx^2