Question

(a) Let X, have a chi-squared distribution with parameter V, and let X, be independent of X, and have a chi-squared distribution with parameter vz. Show that X, + X, has a chi-squared distribution with parameter v, + V Let Y = X1 + Xy. Identify the correct expression for Fly). Fyly) = (f1 +49 (0) BM={{1*(**) 2*3)**- _jei OFW - 1 -{{49)..:@) ib) dx1 FY) = -xq12 dx + -*2/20 1²ax 2 1/2 O Fy(y) :{"(****): 19). x 22 Simplifying Fyly) results in which of the following?

Answer 1

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mG. f of y2 distra is i 1/2 2t).

on MG-f of X, is M, (+) = Vila (1-25) & m.lf of yz is M (t) X2 V2/2 (1-2) then let mof of y * My (t) = Eleth Ele be My(t) tky+X2) Ele tx2) EX txz Ele 2 al Eletxu) C. X & X ane ang independent Mx (t) a My It) (1=2 tj 12 (1-2Ejhhh z 2t j litt qwhich is the M. G. f of a y te diste

> Hence by uniqueness of Mr Gof x, tita ~ V, +17) Y Now 1 Fyly) = P(x + x 2y) [ • YEY ax, +4₂29 P/X, S p z ya XEY*). y you, fo 19Qafy () dx, din, O pdfs option whene Fy (n.) & f (") are Х., -х. respectively. ₂ Hence Comparing we see that 2nd of Fyly) is • a2 YN X2 r Vita where Correct Now Fylyl we'll go fy ludu, )

a ) Xuitur random variable fy (4) is pay of V tuz Therefore, comparing get, the last option (option 5) is the correct form of Fyly). with the options we b) We have already seen that som of independent X² variable with dif u ₂ is x? variable. uitur 2 Using induction x ² + Xvi tast vi El un در ولا teist vastutun) 2.n.(0,1) a 22. x² X tip=1,2,...h independently. a [ 2; 2 wyn z 2 ) Zi h 2 121

2 2 . 21 +24 L (df = 2) x (d.f 3). 2 2 + 22 + 22+23 2 24 -+ 2. L icd 4,61) । 2 ।,2,-.) X D_X.- J/01). 7-12),2,-7n- 2 व 2 J4 7-21, 2) ") 5. 0 (* । 2) (x, म N (difen). n 21 6 n, [rota Notera: E (x² with difn) = D) V M 2n से