Solved: Suppose that Y1, Y2,..., Yn is a random sample from a normal distribution with

Chapter 13, Problem 13.93

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Suppose that Y1, Y2,..., Yn is a random sample from a normal distribution with mean and variance 2. The independence of n i=1(Yi Y )2 and Y can be shown as follows. Define an n n matrix A by A = 1 n 1 n 1 n 1 n 1 n 1 n 1 2 1 2 0 0 0 0 1 2 3 1 2 3 2 2 3 0 0 0 . . . . . . . . . . . . . . . . . . . . . 1 (n 1)n 1 (n 1)n 1 (n 1)n (n 1) (n 1)n and notice that A A = I, the identity matrix. Then, n i=1 Y 2 i = Y Y = Y A AY, where Y is the vector of Yi values. a Show that AY = Y n U1 U2 . . . Un1 , where U1, U2,..., Un1 are linear functions of Y1, Y2,..., Yn . Thus, n i=1 Y 2 i = nY 2 +n1 i=1 U2 i . b Show that the linear functions Y n, U1, U2,..., Un1 are pairwise orthogonal and hence independent under the normality assumption. (See Exercise 5.130.) c Show that n i=1 (Yi Y ) 2 = n1 i=1 U2 i and conclude that this quantity is independent of Y .d Using the results of part (c), show that n i=1(Yi Y )2 2 = (n 1)S2 2 has a 2 distribution with (n 1) df