Question

Solve the problem with all necessary steps in detail.

30 points) Let X1, X2, ..., Xybe independent, identically distributed random variables with p.d.f. f(x) = 22,0 sxso. a. Let Yn be the maximum value of the sample. Is this an unbiased estimator for @? If not, find a constant c so that co is an unbiased estimator. b. Calculate (0) and the Cramer-Rao lower bound for the variance of an unbiased estimator for e. C. Find the variance of the unbiased estimator you found in part a. Compare it to the lower bound you found in part b. Comment.

Answer 1

soluten The provided informatrn in the questry let x, X2 --- xn be independant, identically distributed sandam Vortables with polet - tex) = 200 olx Lo 02 a) let In be the of sample maximum value I d the sample ile yna max X 1X2-2.xn-) To tind pidet of in D(n) om alimni [102]*Cl-ECBC) ] "", fico) flyg) = n CF(x)] for). How to find Fore) F(x) = pe toode - I are dre = 2 ( 27 Hyn)an find 99: 28 OCBC @ - an gen-1 _Synce 02n togo 2n-1 san fon Olyn_Q 2n

trgn = 20 yen olyn.Le -Elyn) og tegn-dyn Coan 1 zny 20-1 on din you?nt 20-_- - 2n oan coan I ntl en roznti 7 anti ( want ELY») zny (03231 7 com 1 In is not unbiased estimator of a P2nti ant 2n. 25 - Elant yn) =' znati Eiyn) . El2ntl yn - 2nt an ce hence c ô B unbiased estimator of ce where = anal & Ogn- 20 on 2n 2 oti" 2n b) to calculate Ico) I [(O) = E(-2²logt (0) 7 102 -5432109412i10) - = E(=3² 2 2 logge inlogie] Ilm) ==EE 1-- ECU to the

ECX)= Jose fix)doc 002 25 0 1 3.6? Jo 2 Las TE(X)= 2.0) E(2x)= 0 T = 32 UCT) (WCT)) (cramer fato lower bound) ni Ilo) 2 vorrance of an unbrated estimator of a @ hxo a vet), 3.02 et) > 202 c) T. anthryn vet) Remig? vc-yn) How to find Ecgn) = Eign) - Elgn) to bind Ely?) = Jy blyn)- a y 2 an yang dan ry2 =an ozn yan+ 2 yo = 20 antota zhtge

ean S N(T) = (2nt reyn) = fant j [ Ecyos) -LE(Ym)7?J = true frente a la h or] men - znalan (znal) ².2n² (2n+2) / ent2) (anti) WCT) = /&nal) ² - (2n+2) 2/02 (2n+2) (anti) gi o v I by competing fo minimum vorrance is bout " 2 .2