Q2. (a) Prove Theorem 9 of the book. (Hint: Use moment generating function (m.g.f.)). (b) Prove Theorem 10 of the book. (Hint: Use moment generating function (m.g.f.)). (c) Prove the identity E(X -...

Extracted text: Q2. (a) Prove Theorem 9 of the book. (Hint: Use moment generating function (m.g.f.)). (b) Prove Theorem 10 of the book. (Hint: Use moment generating function (m.g.f.)). (c) Prove the identity E(X - H)? = E,(X; – X)² + n (X – H)?.


Extracted text: THEOREM 9. If X1. X2,...,X, are independent random variables having chi-square distributions with vi, v2,..., Vn degrees of freedom, then Y = EX; i=1 has the chi-square distribution with vi + v2+ +n degrees of freedom. THEOREM 10. If X and X2 are independent random variables, X1 has a chi-square distribution with vi degrees of freedom, and X1+ X2 has a chi-square distribution with v> v degrees of freedom, then X2 has a chi-square distribution with v-v degrees of freedom.