Question: 3. (30) Let X - NO. E), where E is an n x n positive definite matrix. Assume that XAX follows a x' distribution, where

3. (30) Let X - NO. E), where E is an n x n positive definite matrix. Assume that XAX follows a x' distribution, where A is symmetric matrix. (a) (10) Show that X'(2-1 - A)X follows a x distribution. Assume FixRN 7" distribution and Ais Symmetric mate X ' I E - - ATK - X'EA - X'AX - 9 (b) (10) Show that X"AX and X"(2-1 - A)X are independent. FAX and K'CE - All are independent if and. if AVB = 0 and it is ideaparent A ELE ". A) : AEE- . AEAND [5"= [ 3 A -AEA . since I is gearlived invers of A, So, ASAER is idenpiment " XAX and *15- - All are independent. (c) (10) Find a constant c that may depend on A and E such that cX"(E-1- A)X/(XAX) follows an F distribution

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