Introduction to Probability and Statistics for Engineers and Scientists

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