Question

Proof this

Theorem 2.1 Let X1,...,X, be a random sample from a population with mean J and variance o? < 2. Then (i) E(7) = j, var(7) = ?, and E(S2) = 02. (ii) The moment generating function (m.g.f.) of X is Mg(t) = [My(t/n)]”, where My(t) = E(e*X) is the m.g.f. of X.

Answer 1

Based on the given information,

Given Xi,2, *h populahon a randon samaple from mith mean u and vauâme 62 is Proof: 2. -(x-M)² 262 fea). थिा

Byn definition of mean, E(X) - ja)de fcx) da (a-A 262 da Put y =ブu) dy = da Substitnting, we get 26 dy ( ytM/: M. paf of the distö bution -4/2 + MC) to

(Also, since the firast part 4 the above š an integral zeu) on add frnetion, the value of it ir 0+M E(X) M : VCx) - 62 To prove Byn defmihen of variamce, VCx) = E(X-pg da Lit x-=4 jem 262 da 00 2. al u - dy Let and ye dy -Jrdu fudu Sote in

*Correction in the above image : $dv=ye^{-y^{2}/2\sigma ^{2}}dy$

becomes -y%/20- 6. े when to ptf of distosbution o+ 6?(1) prove" where is -the fot X Prool. Lising the perpertyn of yl, fs s hudegoadurit varniables x and y independurit () M (t) F() Σπή -t

mdepantent oatom vamables, (x+ (x) (M,6") x Xa.)י כי Kp. 2. woith mgf E(e") +(x) every xi has same (that of nomal wible)