Question

10) Suppose that X follows a chi-square distribution with m degrees of freedom and S=X+Y. Given that S follows a chi-square distribution with m+n degrees of freedom, and X and Y are independent, show that y follows a chi-square distribution with n degrees of freedom.

Answer 1

Ans) By the given data, 1 x follows chi-square distribution with m degrees of freedom and sexty. s follows chi-square distribution with (mto) degrees of freedom. .: Mary) (t) = Mx Ct). Myle) . (1-26) inth = {(1-24) ] Melt) (1+2+)?. (1-26) 2 = (-20) 1M, Lt) T-2+)7 - My 1177 The above, is the moment generating fonction of chi-square distribution with in' degrees of freedom. .: Hence proved that y follows a chi-square distribution with a. degrees of freedom. 2