STAT 514 - HW 1 Solution Key

# STAT 514 - HW 1 Solution Key - 3 w 91 Let 3(3 82 Since...

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Unformatted text preview: . _ 3.; w ' 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” only-van . 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 ~ IXa-nl>‘l~li5+‘r‘|) g P(|Xn-a|> 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 aan-woodincexnwaoinprobability. 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 VofX-n 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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