Section 7.3 (p.392)

 

1.         Compute AA’ and A’A to see if they are both identity matrix I. You can show that (a), (b) and (e) are orthogonal, (c) and (d) are not.

 

2.         (a) Let A = . For A to be orthogonal we have AA’ = A’A = I. Therefore, we have the equations

Solve them we get  or

 

            (b) Let A = . Then AA’ = A’A = I.

Focusing on a’s, we get equations

There are infinitely many solutions to the above equations. We can choose, for example, .

For b’s we have

Solve them using the obtained a’s we can get  as a set of possible answers. Therefore, we show that one of the possible orthogonal matrixes is

 

3.         Let Y = (Y1, Y2, Y3) and X = (X1, X2, X3). Then Y = AX where A is

It can be shown that A is orthogonal. By Theorem 2, Y1, Y2, Y3 are i.i.d. standard normal distributions.

 

5.         (a) Since (Xi - m)/s has a standard normal distribution, Y = S (Xi - m)2/s2 has a c2 distribution with n=16 degrees of freedom. The asked probability = Pr(n/2 £ Y £ 2n) = Pr(8 £ Y £ 32) ».99 - .05 = .94 (from table on p.690).

 

            (b) By Theorem 1.  has a c2 distribution with n-1=15 degrees of freedom. The asked probability = Pr(n/2 £ W £ 2n) = Pr(8 £ W £ 32) ».995 - .075 = .92 (from table on p.690).