Question

How to do (d) and (e)? Thanks.

11. Let X, X1, X2, ... be independent and identically distributed random variables taking values 0, 1, 2 with px(0) = 1, px(1) = 3 and px(2) = 1. Define Sn X1 Xn, n > 1. (a) Compute the probability generating function of X (b) Find the probability generating function of Sp. 2) from the probability generating function (c) Find P(Sn (d) Derive the moment generating function of S from its probability generating function and then compute E(Sn) and V(Sn) |(e) Use the moment generating function method to prove that S-E(S,) d, z V(Sn) where Z N (0, 1) with the moment generating function Mz(t) = e, te R. 2+2+2+3+5=14 marks] (a) Px(z) e(I+) = (b) Ps ()(z+1)2n n(2n-1) (c) P(S, 2) 4n (d) Ms, (t)(e + 1)2n, E(S,) n(n + 0.5) and V(S,) (e) A correct argument for Ms,-ES (t) approaching er pected as n oo is ex- VV(Sn

Answer 1

No -O Sdution Fndaperdert ahe idtiany ard randon Vajables, distoibud taking valuas --- +X a) Peotab grating functon af Gtz= X-D I + 7 + z" 4 Б) ржbаьi ity gerwatig function 4 Sn E (25)El)E).... E() E ()n AS х, х, X- - - Rardom vaables 2n

Poge coeticunt af2 in Gsylz") NO- P(Sn2) CoEfident 2nx (2n-1) n(2n- P(Sn 2) d) function af Sn gererating Homent HStEln) Fam (b) Gsnlet) (le' 2n t E (Sn dt dt in- - (2n ) 2. ten 2n- ( 2. 2n- E(Sn) 2nf n ECSn n

Poge NO-3 ELS db dt d 2n ttl dt d dt 21 t t-0 d(n) )2 - e dt Lt (2net ELS t-0 n-L Cn) (2n-) 4 on 2n 2 nn 2 2 n(n-2n 2. 200+2n 2n tn El ntonti) E (SES vis n2nti -(n) 2- 2ntn-2nt 2- 2-

Paga - ON Sn- E(S) Let Snn Ele Sn-ElS) )-. A) t e E Ele F n! St Ell - his)

ON But EL E(x) - PCx-6).0 Pa-1).J + p(xe2).2-1 2 Q= |- %.T tT 2- E(- (2-1) tS Shts) 4 t 22t)). 2n n clarly Shis) n! hs) 90

Poge NO (6 20 oJ tae (E) AS Sn-ElS shere Sn-E(Sn) d shere NLo,Lith gineating function Moment teR