Question

3, Let X, X2,X, be independent random variables such that Xi~N(?) a. Find the distribution of Y= a1X1+azX2+ i.(Hint: The MGF of Xi is Mx, (t) et+(1/2)t) + anXn +b where a, 0 for at least one b. Assume = 2 =n= u and of- a= (X-)/(0/n) ? Explain. a. What is the distribution of The Sqve o tubat num c. What is the distribution of [(X-4)/(0/Vm? Explain.

Answer 1

Question Kw) x x N/A , ); t= 1, 2, - ,h - X1, X2, - - , xn are independent (a) As x, ~ N/M, 6,?), Mott! 6t2 meg. f of Xi , Mxi(t) = 0 as X, v N1M2, 02?) - MX (t) = e e M₂ t + £ 6²+2 MX₂ (t) = :=- = - Xu v N (Mm, on ²). As Unt + 12 on ²t² MXnIt) = Sanice, Y= 0, X1 + Os XAF -- + Ons Xy + b Minif ofy is obtained as, Mylt) = Ma, xot@X+--- t am Xntb (*) = Marx, ft). Maxelt) -- - - Manxolt).ebt [sunice X1, X3, --, Im are vidependent] = Mx, la,t). Mx, (at) --- Myy lant). ebt Mila, t)+4 0;"(0, +)? M7(+)+LoD (93+)? Men (ant) + 4 (ant)?. bt

(674), art Me Ayat-- + 4han) t+(a?eta?.*-- 60762).t? Mylt= ě Thus Y~ N(b+, M, 476b2+ -- Fan Men, azota26 trupation) Xn are a. ld. Nl4,02 (6) X2, X₂, x = 1 x x~ N(M, oh) 7- () ~ N(0,1) - Answer Mx, (+)- Mxs /t)= --- = Mxult); e MA + / 02 ce Malt) = Max=24(t) = MF-u) 58 (+) – Mag-u) (+ sn) Letra de M5 (tm) in é LD..[M3 (4F)]

Thus, Malt)= é en =é domy.. seulement latem) 1 [using o] - Asn.u tonu + Mz1t) = en which is mig.f. of N/0,1). Thus z= (x - 1) ~ N/0,1) lolin / - Answer Answer - (c) sunce z= (x - 1) ~ N(0,1) then z=(74)*~XC = Answer -Olaco pidid of 2, 81x)= t.ex; -oczkos Mhf ofz? Mz2(t)= Eletz) flzoda

Melt) fortsette et de ſé (24)zdz. (32+) z w = (3) W - dz= √ 1 t at w dw Mz2(+) = lén. J eta.dw thus - Vk : () ed - Ma Ta se veshjet M.af. of 2, M22 (t) = (1-2) ² Thus 2² (7-4) ~X (1) Answol