Fundamentals of Probability and Statistics for Engineers

340 Fundamentals of Probability and Statistics for EngineersThe least-square estimate of q, ^q, is found by minimizing Q. Applying thevariational principle discussed in Section 9.3.1.1, we haveSetting dQ ˆ 0, the solution for ^q is obtained from normal equationorwhich givesdQ ˆ dq T C T …y Cq† …y Cq† T Cdqˆ 2dq T C T …y Cq†:C T …y C^q† ˆ0; …11:14†C T C^q ˆ C T y;^q ˆ…C T C† 1 C T y:…11:15†In the above, the inverse of matrix C T C exists if there are at least two distinctvalues of x i represented in the sample.We can easily check that Equation (11.15) is identical to Equations (11.7)and (11.8) by noting that2 3andC T C ˆ 1 1 1 x 1 x 2 x nC T y ˆ 1 1 1x 1 x 2 x n2641 x 121 x 2n6 .4. 75 ˆ 4nx1 x n32.7. 5 ˆ 4y 1y 2y nP niˆ13ny5;x i y inxPnx 2 iiˆ135;2 3 123n nx ny^q ˆ…C T C† 1 C T y ˆ 4nxPnx 2 5 4 P n 5i x i y iiˆ1iˆ12 P ynP x i y i nxyn1 3x 2 inx 2 xiˆ1iˆ1ˆ P 6 nP x i y i nxyn1745x 2 i nx 2iˆ1iˆ123y ^x6ˆ P nP …x i x†…y i y†n1 74…x i x† 2 5:iˆ1iˆ1TLFeBOOK