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hw7_sol - aauemi{KW-=0 0.3 FLXEL 1.10.2.— W‘...

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Unformatted text preview: aauemi {KW-=0: 0.3 FLXEL) 1.10.2.— W‘: PU=8)-=—=P Pflxto):3f> o‘B+o.L+-i>+3p ==~l :7 P: 9......5— :. 042.9" L1 ... D‘PCX:&)-}- ; ~P£Xul>+L.PCX=3-> +3.PC‘K'=3) = 035+ 0M -+ 0-373" = LD75’ Nah. 4% Eth is V1042 kw 01.5%. pmblfimz— PCH)=P , PLTD= PP- 1112' \K be 4%}: Random VMJ'A‘b/er dflwafil‘mg +112. hUw'LbU-r :93], 39-13;)5 {AM-HI eff-hen... hmix m +MI‘IK has 0:.ch Mtg ~ ”NJ 5:» 75415 Ct?“ occm hung, m 4-14.21 acoltuwma swmw (hf fly); on? X W Pwmb. 1.43 2- H, H (P f“ 1- "T, T Q- P) 3 H: T; H 1-0—19) 3 H )T; T— ECI—P)L 5 T: HLE‘ (PP) P1” 3 T: HM PUFF)?" PLX:;): P’C‘iH) H})+ P({T}TB) =~ F1+ (VF); g {Fa—l #2.? +p’” PLX=3>= KEHITJHE)+F(iH;T;TE)+P(i—r;HJH3>+P<1TEHJTE) -=- FLU—PH ;>(a-—P)“+ F‘Ct-f) + PUFF)” ==— QFCW) +P<t—-P)L] =— all Fl’P3+ PEFLPWP] = .2 ( P— F“) Random Vatwtxhuew 11 4) ELK] = E ( I: ) pk(1 — p)“‘kk Note it = 3 term does not k=fl contribute to sum. 11. Z :kr—(nfl _ k)!p"‘(1 _ flunk i=1 _ (fl. _ 1)'k—1n—k r _ _ up :1——_(k—1]I(n—k)'p 1(1— p) letk _Ic—1 _ “—1 (n — 1)! k’ n—l—ia’ — HP a; m? (1—?) sum of Binomial probe with parameter n — 1 and p = up a! i flu Ein] = Z flkflmpk (1 — p)“"‘ cancelling I: and letting ic’ = is: — 1 “E1 51*]. I ‘ :PZ(k’+1)( k; )P"(1-p)”'1"‘ H=D II a at M1 H 3": affix E! 3:] l—I. H..___..J' "u i"; f-‘x i—I. '1': H-u—F' :2 I H |' i1" + M” | H fl 3 E15:1] 1...: Nah—J 36% r—fi l—1' fl H—pn' :3 I H- “I. expected value of Binemial 1 with parameters in — 1 and 11 WW -1)p+1} Hi: = 5W] —5[X]2 = ”PH” —1)P‘+1“—”P} = ”PU -P) 5) 5X: m(#(_1)) __ m1 m I] j; 1 1 a: rim—[1 $d$:lfl$1 am 1 W .1: fl) £[X] = [mifil + mad: +1: «(1 + I“)d$ Consider the latter term: If m _ 1 3 "' __ 111(1 + y) In 'rr(l + mfidm 2w 111(1 +2: )5; _... 2n —+ no 3) a) Net reward Y = 1::le2 +bX—nd, where X is a. binomial RV E[Y] = £151le + bE[X] — :16 = «(EEIXI + Var [X]] + bE[X] — nd = quip: + npq) + Imp — nd b)}’=n’" = U+L).€—s£ 5[Nu: Mail: 2;“) EENM: ’Wt/u, ,] {an Problem lfl Gt) 0 J 1&9- ati>= .l--C1:-~l.,)) mix-Eb 5) Lot 701(19195 Ham-r flan-$11? qun¢£:r’m 51 fiNaAMAé 593/. w:‘% men/u “at! firming. 0-5: M ffxbc) dx + I 392(1) d1. = I .30 )1» FMWI 3y MM&+X.? ear-f. 332(1) )1 jfx(3rv)obfi L'- J; “”00 a mfg 1%,“ =7 Mel-GUEM' = /Ul ...
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