Fundamentals of Probability and Statistics for Engineers

The least-square estimate of is found by minimizing Q. Applying the
variational principle discussed in Section 9.3.1.1, we have

Setting the solution for is obtained from normal equation

or

which gives

In the above, the inverse of matrix CTC exists if there are at least two distinct
values of xi represented in the sample.
We can easily check that Equation (11.15) is identical to Equations (11.7)
and (11.8) by noting that

and

340 Fundamentals of Probability and Statistics for Engineers

q,^q,

dQˆdqTCT…yCq†…yCq†TCdq

ˆ 2 dqTCT…yCq†:

dQˆ0, ^q

CT…yC^q†ˆ 0 ; … 11 : 14 †

CTC^qˆCTy;

^qˆ…CTC†^1 CTy: … 11 : 15 †

CTCˆ

11

1

x 1 x 2 xn



1 x 1

1 x 2

... ...

1 xn

2 6 6 6 6 4

3 7 7 7 7 5

ˆ

nnx

nx

Pn

iˆ 1

x^2 i

2

4

3

(^5) ;
CTyˆ

11

1

x 1 x 2 xn



y 1

y 2

...

yn

2 6 6 6 6 4

3 7 7 7 7 5

ˆ

ny

Pn

iˆ 1

xiyi

2

4

3

(^5) ;
^qˆ…CTC†^1 CTyˆ
nnx
nx
Pn
iˆ 1
x^2 i
2
4
3
5
 (^1) ny
Pn
iˆ 1
xiyi
2
4
3
5
ˆ
y
Pn
iˆ 1
xiyinxy

Pn
iˆ 1
x^2 inx^2
 1
x
Pn
iˆ 1
xiyinxy

Pn
iˆ 1
x^2 inx^2
 1
2
(^66)
(^64)
3
(^77)
(^75)
ˆ
y ^x
Pn
iˆ 1
…xix†…yiy†

Pn
iˆ 1
…xix†^2
 1
2
(^64)
3
(^75) :