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91': Let 3(3):: 82. Since 5(5) :3 a. convex function, we know from Jensen’ 3 Inequality that 115(5)
5.11 " > 5(E3), which implies 02: E532 > (ES)? Taking square roots, 0) Es. From the
proof of Jensen’ 3 Inequality, it' as clear thatf In fact the inequality will be strict unleu that
ism interval Isuch thatgislinmonlmd P(X€I)=_l. Sinoenzin “linear” onlyvan
. tingle points, we have ET2 > (ET)2 for any random variable Thanks P(T=ET) = l. 5,]? Let Xi 2 weight of ith booklet 1:: package. The xi 3 are iid with EXi z 1 and Var Xi“ .. H052
520:“ We want to approximate P( 2.109 11:1 > 100.4): P( 2 .12? Xi/IDO > 1.004) =_P(X 2,» 1.004).
By the CLT, P(X > 1.004) m P(Z > (1. 004 1)/(. 05/10)): P(Z > .8) =: .2119. .5/{5 a) Foranyé>0y  ...\ :32. Elm—«aw: Im—EIIWWINIWWD
~ IXanl>‘l~li5+‘r‘) g P(Xna> NS) w» 0, an w» 00, since Xn —. e in probability. Thus K; r «E in probabiiity. b) For any e > 0, ﬂew—05:9. +02 .
”(MT$75M”??? (a+—<a+f~‘_;) "P([Xn—n<lu w)—rl aanwoodincexnwaoinprobability. Thus o/Xn ..... i in'probability.
c) 5% _. ,2 in 13,053,133]th By 1:), Sn = R .4 r r. or in probability. By b), a/Sn : I in
probability. ﬁt! Using EYB— .... —p and VofXn z 172/13, we obtain 5‘31  Eﬂx ”‘0 ‘Lﬁmxn =.u) ﬁrm—*0
150‘ n—I‘)_ nv 2 Ya: (Xnup) =%Varf=f§%»m 0' “MI: ...
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 Spring '07
 SENTURK,DAMLA
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