Sloution to problem set 4

# Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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Unformatted text preview: S|o4 EEEEEEEEEEEI 78. Haysth works) : Pt I - 2 works LJ 3 -- 4 works] 2 P11 1 w 2 works} + P{ 3 — 4 works} - P( I -— 2 workxﬂ 3 -— 4 works) 2 P(_l wurksu 2 works} + PC! works ﬂ 4 works) — P( I — 2 ) 0 PIS — 4} =( .9+.9-.81] + (HHS) '- (.9+.9-.81)(.9)(.9} : .99 + .81 - .8019 2 99%| 83. a. Let D. : detection on ISI ﬁxaliun. D3 : dctectinn on 2IHI ﬁxation. P(dctecti0n in at most 2 ﬁxations) : PILDIJ+ P'fo (“a D1) : 11ml ) + P(D2|DI')P(D1) =P+P£l "P}=P(2"P3- h. DBﬁI‘IE 13., D3, .Dn as; in 3. Then P{:1t moat n fixatiuns} =P{Dl)+P(D.’ r‘“. D3} +P{Dl’ (‘1 D3’ n D3)+ ...+ P{D]’ n D1’ (1 (‘1 D....’ n DI.) =p+p{l --p]+pn[l--}:I)'+...+p(l--}:I)“'I 1—(1—p)” 1—(1- p) Altcrnalivcly. P(at must n ﬁxaliuns) : l - P{e1t least n+l are req'd} =p[l+[l -p}+{| -p)1+...+(I—p)h'1]: p! =]_(1_p)ﬂ = l — Pﬁm detection in l5l n l’ixations) = l — P(D.' ﬂDl’ ﬂ nDn’) I l _ U m P)” c. P{no detcction in 3 ﬁxatilms} : {I — [1)] d. Ptpasses inspection] 2 PHnnt ﬂawed} LJ {ﬂawed and passesh : P{1wtf]awed}+ Pt'ﬂawed and passes} = .9 + Ptpasses | flawed}l P(f1awed) : .9+(1 n pf“) . _ 3 e. PU'lawed Ipassed} = W = Firm-9d) .9 + .1(1— p)- 3 Fur p = .5. Ptﬂnwed I passed} : ' 1L5) = _0137 .9+.1(.5}3 When three experiments are performed. there are 3 different ways in which detectinn can occur on exactly 2 of the experiments: {i} #1 and #2 and not #3 {ii} #I and not #2 anti #3: (iii) nnt#l and #2 and #3. If the impurity is present, the probability of exactly 2 deteetinns in three (independent) experiments is (.3)(.8)[.2J + {.3]{.2](.3] + (.2}(.8){.8} = .384. If the impurity is absent. the analogous probability is 3(.l}(.l )(9) = .027. Thus Ptpresent I detected in exactly 2 out 01—3} 2 Pidet ecfed.in.exaeﬂy.2 r“. present) P(det ected.in.exnctiy.2] (.384)(.4) : —= .905 (.334)(.4) + (.027)(.6} Ill-1. .15 .5 WHERE i .0525 R1~=CR3~=CR2 // .15 .25 G R3421 IDSFS f5 .El?5 El 1-:R2-cR3 .15 .525 .1 5 I I I 3. PH] I R. n: R: q RILI = — = .ﬁ? . F{B | R. {R3 n: RI1]=.33. dummy ab granule. _ IS + .075 .UﬁZS h. Pit] IR. (RI. c R1] : = .E‘MI c .ﬂﬁ. 5L] cluxsit'y up. hasﬂl. .2 I25 JETS PM I R_. c R. < R1: = = .ﬂﬁﬁ? . 5L1 clmi r3; in; basalt. .56125 c. Ptrrmllcum Elam-1isz P113 L'lamaii' i-J..‘i G: + P“: clzmsaif :15 B] : Piclaﬁhifax U- f BJPfBj + Htlamii' 3:; E- | ﬁll-7(5) ZHRIiﬂiikaIBH.?35+P1R1{R;{R;UIR4; {R-I (R; IGJLESI = [.HHLTSI + [.25 + .lﬁHlSII = .I?5 :1. Fur wins: values; of pwil] P16 |R.-=:R_.::R_1j::- .5. PH] ER. a: R_». : R112:- :1 P15!R_.~:R. q R1] :- j'.’ .51 .15 I Ham. {R3 cm]: —p=—P:-.5 iﬂ' p 2:— .ﬁp+.l£1-pj .I+.5p T .25}: __ 4 BEER. {Rd {R3}: —::_5 .H p :— _25p+_2{1—p} 9 Jim _. I4 . . FIBER. {R.n:Rl1= — 3:- .5 M p }— 1muslm51nclwe] .l5p+.TII.'i-p,1 I? I4 II' p 3? E aha-aya- ﬁnally as granile. HIT. . I | a. Pliallmcurrfctrmm}: —=—=.ﬂ*4|7" 4X3X2>¢l 24 h. The ‘i uulcumm which grit-id Luann-cut assignnwnh are; 2143. 234 t. 1-4 | 3. 3] 42, 3412. El 342L4l11432Land43|15u Frau incurred] : — 2.315 24 Section 3.2 l]. I1. [LS 0.4 I13 0.2 III Ill] 'I-I.I"\r'| -'|u'l -_1mr. a. L'. Pix =IEI‘J =.4EJ+_I_"1 || . Lil 'JI P13}f31=.|5 Section 3.3 2.8. J- a. E{X]-= Ex r mix) .I.=|::' =[ﬂ}||:.[}3J+ {1][.|5}+ [2]{.45}+{3][.2ﬂ +{4Hﬂ5j = 2.11“! J- I1. WK}: £{x-2ﬂﬁf ~p{x} =m— :_ﬂﬁf[_tam+__.+{4— 1&3me .1.=|!!| = .339133+.1ﬁ354ﬂ+_[ﬂ1ﬁEfH.2335?2+_lﬂﬁlxﬂ = 93154 :3. 5154.936 235?? .|.'= I] .1 d. mm = x: - pm] — (215633 = 5_|3m— 4243:; = .9354 lﬁ. a. x ﬂutmmes put} {1' FFFF { .7" ‘14 =24ﬂ1 l FH—‘STIII-‘SIa'_F51a'I:.5L-'1=IJ 4|+_J:f':_3:|] =41 1:3 2 I"FS S,l"51"S.SF1"S.l-'S H FEES I‘.SSI"I" {1Hin .3 II: I =2ﬁ4ﬁ 3 F55; 5;. SIHSESFSSSSIJ 4| [.THJ r" | =.m‘5& 4 3355; 1.334 =mm I1. .IIL‘I 55::- 54 £5 .20 .EE- c. - CI E 2 3 4 III 5 area c. [)[M '15 largest I'm X = I d. Pix 2 2] ‘—- p£31+ pi?“ + pH) 3 .Eﬁiﬁ-F.EJTI'Sﬁ+_[H}EI : "1433 This L'mlld 31er Inc dun: using 'th' untilplcmunl. Section 3.3 2.8. J- a. E{X]-= Ex r mix) .I.=|::' =[ﬂ}||:.[}3J+ {1][.|5}+ [2]{.45}+{3][.2ﬂ +{4Hﬂ5j = 2.11“! J- I1. WK}: £{x-2ﬂﬁf ~p{x} =m— :_ﬂﬁf[_tam+__.+{4— 1&3me .1.=|!!| = .339133+.1ﬁ354ﬂ+_[ﬂ1ﬁEfH.2335?2+_lﬂﬁlxﬂ = 93154 :3. 5154.936 235?? .|.'= I] .1 d. mm = x: - pm] — (215633 = 5_|3m— 4243:; = .9354 ...
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