ᆞٳ֥҃ႋႨ 1.ਬࡱ֥ܿ۬ഡ࠹ ႮሱളӁཌࡆ۽֥ଖᇕਬࡱ֥ଽࣥXčݸĎڛՖᆞٳ҃N(µ,1)đनଽࣥµ൞2րק֥đॖၛ๙ݖטᆜھሱളӁཌটഡקđٚҵσ=1ّ႘ᆃ่ሱളӁཌ֥ࡆ۽ࣚ؇bೂݔࡆ۽֥ਬࡱଽࣥཬႿ10ࠇնႿ12नູ҂ކ۬đఃჅູކ۬bཧ൲ૄࡱކ۬ࠆ০đཧ൲ૄࡱ҂ކ۬đၘᆩཧ൲০Lčֆ໊ğჭĎაཧ൲ਬࡱ֥ଽࣥXႵೂ༯ܱ༢ğ ⎧−1X<10⎪L=2010≤X≤12 ⎨⎪−5X>12⎩໙ğनᆰࣥµູޅᆴൈđҌିཧ൲၂۱ਬࡱ֥न০ቋնĤ ႮႿL൞ෛࠏэਈđ൞X֥ݦඔđ෮ၛन০ູࠧ௹ຬ০bႮX~N(µ,1)đପહX−µ~N(0,1) E(L)=20P{10≤X≤12}−P(X<10}−5P{X>12} 20P{10−µ≤X−µ≤12−µ}−P(X−µ<10−µ}−5P{X−µ>12−µ =20Φ(12−µ)−20Φ(10−µ)−Φ(10−µ)−5+5Φ(12−µ) =25Φ(12−µ)−21Φ(10−µ)−5 ॖᆩđ௹ຬ০აनଽࣥµႵܱđ൞µ֥၂ჭݦඔb dE(L)ູਔ௹ຬ০E(L)֥ቋնᆴđ=21ϕ(10−µ)−25ϕ(12−µ)=0đఃᇏdµΦ(x)aϕ(x)ٳљູѓሙᆞٳ֥҃ٳ҃ݦඔაۀੱૡ؇ݦඔđᄵ 2222(10−µ)(12−µ)(10−µ)(12−µ)−−−−21252222 e=e ࠧ 21e=25e 2π2πࢳᆭđ֤ 125 µ=11−ln≈ 221ႮՎॖᆩđ֒नଽࣥµഡקູݸൈđॖཧ൲ૄ۱ਬࡱ֥न০ቋնb 2 ႋھܓઙྍЇልࠏ ऴ٥ӌളӁ၂Ѐᇗ֥ܷልऴ٥đሱЇልཌഈնਈඔऌіૼđૄܷᇗਈ൞ڛՖѓሙҵູ Ѐ֥ᆞٳ҃bູਔૄܷऴ٥ഒႿ Ѐ֥Ӂ҂؟Ⴟ đႋϜሱЇልཌ॥ᇅ֥नᆴµטࢫ֞હ໊ᇂഈĤ၂ྍ֥Їልࠏࡎ۬൞ ຣჭđЇል֥ऴ٥֥ᇗਈڛՖѓሙҵ Ѐ֥ᆞٳ҃đဢູਔૄܷऴ٥ഒႿ Ѐ֥Ӂ҂؟Ⴟ đႋϜሱЇልཌ॥ᇅ֥नᆴµטࢫ֞હ໊ᇂഈĤ 2ഡXіൕჰሱЇልཌഈ၂ܷऴ٥֥ᇗਈđᄵX~N(µ,)đࡌೂϜሱЇልཌ֥न2ᆴµ॥ᇅᄝ Ѐ໊֥ᇂഈđପહႮႿX~N(1,)đᄵഒႿ Ѐ֥ऴ٥ေᅝಆ҆ऴ٥֥ đࠧP{X<1}=đᆃ൞҂ކေ֥b
ູਔഒႿ Ѐ֥ऴ٥෮ᅝ֥б২҂؟Ⴟ đႋϜሱЇልཌ֥नᆴµ॥ᇅᄝб Ѐն໊֥ᇂഈđఃᇏµсྶડቀۀੱٚӱP{X<1}=ࠧ X−µ1−µ1−µ⎧⎫⎛⎞ P{X<1}=P<=Φ= ⎜⎟⎨⎬⎩⎭⎝⎠µ−1µ−1⎛⎞Ⴟ൞ Φ=đႮՎॖ֤=đՖط µ=ࠧϜሱЇልࠏ֥नᆴטࢫ⎜⎟⎝⎠֞ ໊֥ᇂഈҌିЌᆣഒႿ Ѐ֥ऴ٥҂؟Ⴟ bࠧğनૄૄܷေ؟ል Ѐb ೂݔྍઙ၂ሱЇልࠏđྍࡹ၂่ሱЇልཌđྍЇልཌഈૄܷऴ٥֥ᇗਈູYđᄵ2Y~N(µ,)đູਔഒႿ Ѐ֥ऴ٥෮ᅝ֥б২҂؟Ⴟ đႋϜሱЇልཌ֥नᆴµ11॥ᇅᄝб Ѐն໊֥ᇂഈđఃᇏµсྶડቀۀੱٚӱP{Y<1}=ࠧ 1Y−µ1−µ1−µ⎧⎫⎛⎞111 P{Y<1}=P<=Φ⎜⎟= ⎨⎬⎩⎭⎝⎠µ−1µ−1⎛⎞11Ⴟ൞Φ⎜⎟=đႮՎॖ֤ğ=đ෮ၛµ=ࠧϜሱЇልࠏ֥⎝⎠नᆴטࢫ֞ ໊֥ᇂഈࣼିЌᆣഒႿ Ѐ֥ऴ٥҂؟Ⴟ bᆃဢनૄܷࠧॖࢫჿऴ٥−=Ѐb ၛૄರॖളӁ ܷऴ٥࠹ෘđᄵૄರࣼॖࢫჿ2000×=192Ѐऴ٥bೂݔૄЀऴ٥֥ӮЧ൞ ჭđᄵ۽ӌૄರॖᄹࡆ০ ჭđ ฿ࣼିሉ߭ӮЧđֻ ฿ࣼॖࠆ࣪০bၹՎđھऴ٥ӌႋھܓઙྍ֥Їልࠏb ႮႿሱཌЇል֥ऴ٥֥ᇗਈX൞ڛՖᆞٳ֥҃đᆞٳ֥҃ٚҵّ႘ਔЇልࠏ֥ࣚ؇đ҂ࣇ႕ཙ֞Ӂ֥ᇉਈđطᇗ႕ཙ֞۽ӌི֥ၭb෮ၛđᄝ၂ུӁ֥ᇉਈ॥ᇅݖӱᇏđ۷ᇗေ֥൞ေ॥ᇅٚҵb 3.ॖࠆ֤ӑӁࢂ֥Ӂਈ ၭ౿ທऎӌልӚࡗሙСൌྛ࠹ࡱӑӁࢂđູՎླေؓളӁקحቔԛܿקb۴ऌၛສ֥2࠹ሧਘॖᆩđ۲۱۽ದૄᄅል֥ӁࡱඔXڛՖᆞٳ҃N(4000,200)bӚࡗᇶ༐ຬ10%֥۽ದࠆ֤ӑӁࢂđପહקحѓሙႋھ൞؟ഒࡱĤࠧ۽ದૄᄅླေል؟ഒࡱӁၛഈҌିࠆ֤ࢂࣁĤ ഡxູקحѓሙđପહP{X≥x}=đᄵP{X<x}=1−P{X≥x}= 0000X−4000x−4000x−4000⎧⎫⎛⎞00 P{X<x}=P<=Φ= ⎜⎟⎨⎬0200200200⎩⎭⎝⎠x−40000Ұіđ֤=đ෮ၛx=4000+×200=4256čࡱĎb္ࣼ൞ඪđ۽ದૄᄅ0200сྶልӁ4256ࡱၛഈҌିࠆ֤ӑӁקحࢂb 4 ౼ٳඔཌࠣॉളՑ֥ಒק ଖఒြሙС๙ݖᅱௗॉ൫ᅱ൬300ᆯ۽đఃᇏᆞൔ۽270đਢൈ۽30ĠБॉ֥ದඔ൞1657ದđॉ൫ડٳ൞400ٳbॉ൫ު֤ᆩđॉ൫नӮࠛµ=166ٳđ360ٳၛഈ֥ۚٳॉള31ದbཬᄝᆃՑॉ൫ᇏ֤256ٳđ໙ିڎФ౼ĤିڎФௗູᆞൔ۽Ĥ ٳ༅ğၹູႵ1657ದҕࡆॉ൫đॖၛಪູॉ൫ӮࠛXڛՖᆞٳ҃đनٳµ=166
22ٳđࠧX~N(166,σ)bႮႿᆞٳ֥҃ٚҵσໃᆩđॖၛ۴ऌ360ٳၛഈ֥ॉള31ದᆃ۱่ࡱಒק༯টbಖުყҩཬ֥Ցđࠧॖ߭ճ෮ิԛ֥໙ีbႮ X−166360−16619431⎧⎫⎛⎞ P{X>360}=P>=1−Φ== ⎜⎟⎨⎬σσσ1657⎩⎭⎝⎠194194⎛⎞ॖ֤Φ=đҰі֤=đࠧσ≈93čॖିၹູॉള֥ඣ࿇൹նđ֝ᇁ⎜⎟σσ⎝⎠2ѓሙҵหљնĎb෮ၛX~N(166,93)đႵ X−166256−166⎧⎫ P{X>256}=P>=1−Φ()=1−=đ ⎨⎬9393⎩⎭ ×1657≈275 ᄝ෮ႵॉളᇏđۚႿཬӮ֥ࠛჿႵ275ದđၹՎཬնჿՑ൞276đᄝ300ᆭభđିФ౼bཬ֥Ցஆᄝ270ᆭުđᆺିФௗູਢൈ۽b 5 ܄܋చӚӚۚ؇֥ಒק ऌඪ܄܋చӚӚ֥ۚ؇൞οӮ୍ଳሰაӚஷ֥ࠏ߶ᄝၛ༯֥ѓሙটഡ࠹֥b2۴ऌ࠹ሧਘđӮ୍ଳሰ֥ദۚXڛՖᆞٳ҃N(168,7)čĎđପહӚ֥ۚ؇ႋھ൞؟ഒĤ ഡӚ֥ۚ؇ႋູađପહႋಒקaఃડቀ P{X≥a}≤ 2ႮႿX~N(168,7)đᄵ X−168a−168a−168⎛⎞⎛⎞ P{X≥a}=1−P{X<a}=1−P<=1−Φ≤ ⎜⎟⎜⎟777⎝⎠⎝⎠a−168⎛⎞ =1−Φ≤ ⎜⎟7⎝⎠a−168a−168⎛⎞ՖطΦ≥đၹՎႵ≥đܣ ⎜⎟77⎝⎠ a≥168+7×=čĎ ႮՎॖᆩđӚ֥ۚ؇ᇀഒႋູb ᇿğ aં၇ऌğ ႋႨѓሙᆞٳ֥҃ٳ҃ݦඔі࠹ෘ၂Ϯᆞٳ҃Ⴕܱ֥ۀੱb aႋႨაܼ ᆞٳ҃ᄝሱ॥ᇅaႪ߄ഡ࠹aЇልࠇࡆ۽ਬࡱ֥ࣚ؇ၛࠣᄝᇉਈܵބ॥ᇅ֩ٚ૫Ⴕሢܼ֥ٗႋႨbᆞٳ֥҃नᆴࣼ൞ሱ॥ᇅ֥ഡקᆴđٚҵࣼ൞ሱ॥ᇅ֥ࣚ؇Ġٚҵᄀཬđࣚ؇ᄀۚđ༢֥ྟିᄀݺbᆞٳ֥҃ႋႨ൞ܼ֥ٗđᆃਙई֥ᆺ൞Ⴕཋ֥ࠫ۱ٚ૫֥ႋႨbਬࡱ֥ܿ۬ഡ࠹൞۴ऌቋႪ߄֥නམഡקࡆ۽ਬࡱ֥ଽࣥĠ֒ྍ֥Їልࠏ֥ࣚ؇ննิ֥ۚ౦ঃ༯ႋھܓઙྍЇልࠏĠႨކ֥ٚمಒקॖࠆ֤ӑӁࢂ֥Ӂਈđ֤ӑӁࢂ֥ࢂحିᄝყק֥࠹߃ଽĠ۴ऌॉ൫ӮࠛڛՖᆞٳ֥҃หׄđႋႨᆞٳ҃ಒק౼ٳඔཌࠣॉള֥նᇁՑĠ۴ऌᆞٳ֥҃หׄഡק܄܋చӚӚۚ؇čࠇᆀ၂ུ܄܋ഡീ֥ 3
ܿ۬Ď֥ಒקb ҕॉ໓ངğ ᐐൂ֩ ۀੱંაඔ࠹<.> ᇏݓ࠹ԛϱഠ 4
