Question

12. let Mx(1) be the moment generating function of X. Show that (a) Mex+o(t) = eMx(at). (b) TX - Normal(), o?) and moment generating function of X is Mx (0) - to'p. Show that the random variable 2 - Normal(0,1) 13. IX. X X . are mutually independent normal random variables with means t o ... and variances o, o,...,0, then prove that X NOEL ?). 14. If Mx(1) be the moment generating function of X. Show that (a) log(Mx (0))... - E[X] (b) Slog(Mx(t)... = V[X]

Answer 1

③ we have my Lt) - Eleta) o malo. Je tu fons de iH i Flax+b) ) a) Maxto = E (e ) - E (atx+bt )=E(atx ot && (eat). E test) - MyCat) est (in It Elect) 16 tb (at). Max+b = e Mx b) Given that z=(x-M) It), in It) - Ele+ (x-1)) "Сх = E(et (x ) ) - X-H e Milo ( + + 1 ) (: Mx It) = earth to)

Mit - Att Ź = eo 0 0 um It) / which is the M.G.F. of Standard Norma variate with Mean o and variance i. Hence Z WN(0,1) (OR) 2 = X-N 7 € (2) = E(X)-N = MA 5 E (Z) -0 N- 16 v (x-1) = 1 (u (x) -A) = 1 banoj 3 VZ (X NNCAIR . ZNN (0,1)