Fundamentals of Probability and Statistics for Engineers

where

and

Proof of Theorem 11.1:estimates and are found by taking partial
derivatives of Q given by Equation (11.6) with respect to and , setting these
derivatives to zero and solving for and H ence, we have

Upon simplifying and setting the above equations to zero, we have the so-called
normal equations:

Their solutions are easily found to be those given by Equations(11.7) and
(11.8).
To ensure that these solutions correspond to the minimum of the sum of
squared residuals, we need to verify that

and

338 Fundamentals of Probability and Statistics for Engineers

xˆ

1

n

Xn

iˆ 1

xi;

yˆ

1

n

Xn

iˆ 1

yi:

^ ^

^,

^.

qQ

q ^

ˆ

Xn

iˆ 1

 2 ‰yi… ^‡ ^xi†Š;

qQ

q ^

ˆ

Xn

iˆ 1

 2 xi‰yi… ^‡ ^xi†Š:

n ^‡nx ^ˆny; … 11 : 9 †

nx ^‡ ^

Xn

iˆ 1

x^2 iˆ

Xn

iˆ 1

xiyi: … 11 : 10 †

q^2 Q

q ^^2

> 0 ;

Dˆ

q^2 Q

q ^^2

q^2 Q

q ^q ^

q^2 Q

q ^q ^

q^2 Q

q ^^2

(^)
(^)

0 ;

^

^