可靠度技术手册二项分布可靠度评估技术.pdf

下载

下载

6积分

8积分

下载

8积分

VIP价：6积分

̙ቦ度Ҧஔ˓̅ ɚධʱб̙ቦ度൙ПҦஔ ుᒿᎌ ᇜഹ ʕശ͏਷ɘɤɧ年ɤɓ˜ɚɤ˚ ̙ቦ度Ҧஔ˓̅ ɚධʱб̙ቦ度൙ПҦஔ ͦ 錄 1 ۃԊ.......................................................................................................................................1 2 ዚ率ʱбत׌.......................................................................................................................1 3 ዚ率參數પ֛.......................................................................................................................3 ̻ѩ࠽ၾᅺ๟ࢨʘᓃપ֛...........................................................................................3 ̻ѩ࠽ʘਜගપ֛.......................................................................................................3 HLP:75 ɚධʱб̙ቦ度൙ПҦஔ i 2003/05/25 ̙ቦ度Ҧஔ˓̅ ɚධʱб̙ቦ度൙ПҦஔ 1 ۃԊ f 2 ዚ率ʱбत׌ ί஢εʈ೻಻༊ʕd若ӊϣ಻༊ேୌΥɨ列৿ணૢ΁j (1). ӊϣ಻༊සϞ兩၇̙ঐ೯͛ٙഐ؈dݔԫ΁೯͛א不೯͛i (2). ༈ԫ΁೯͛ٙዚ率dίӊϣ಻༊ʕ㛬މ੬數i (3). ӊϣ಻༊ί୕ࠇɪѩމ־Ϥ޴ʝዹ立f တԑɪࠑૢ΁ٙԫ΁ה࿴ϓٙҏ列၈މݡр利ҏ列(Bernoulli sequence)f৿ணӊɓϣ಻༊ʕdԫ΁೯͛ٙዚ率މpd不೯͛ٙዚ率މ1−pdۆίਂ了nϣ連ᚃ಻༊ٙݡр利ҏ列ʕdԫ΁A೯͛xϣٙዚ率f(x)މᓙᚃۨዚ率ʱбd̙ڌͪϓj Xn⎛⎞n−xx (1) f(x)=⎜⎟p(1−p),x=0,1,2,Ln,0<n,0≤p≤1X⎜⎟x⎝⎠တԑɪόٙዚ率ʱб၈މɚධʱб(binomial distribution)dɓছাމbin(x;n,p)dɚධʱбϞnʿp兩ࡈ參數dnމЗໄ參數dpމҖ狀參數f ɚධʱбٙ累ጐዚ率Ռ數Bin(x;n,p)މj Bin(x;n,p)=F(x)Xx =f(k) (2) ∑Xk=0xn⎡⎤n−kk=p(1−p)∑⎢⎥kk=0⎣⎦ 數ኪಂૐ࠽ ɚධʱб̻ٙѩ࠽E[X]ၾᜊ異數V[X]ʱйމj (3) E[X]=np (4)V[X]=np(1−p)HLP:75 ɚධʱб̙ቦ度൙ПҦஔ 1 2003/05/25 ̙ቦ度Ҧஔ˓̅ ɚධʱб̙ቦ度൙ПҦஔ त׌౜ࠑ Պۨٙɚධʱбνྡ1הͪf ྡ1jɚධʱбዚ率੗度Ռ數 Ꮠ͜說׼ ɚධʱб੬͜׵ཥᒜٙॹ଱ʱбeཥ༑ٙટᇞ፹Ⴌʱбeዚ૛׌ॹ௘ʱбeٞ౺˪ٙॹ௘數ʘ類ࠇ數ۨ數ኽe̒ኬ᜗ጐ᜗ཥ路ٙ良ۜ率൙Пd˸ʿϓ઻ۨӻ୕̙ٙቦ度אϓ̌ዚ率ഃf̤̮dɚධʱбɰ੬͜׵ࠇၑልᕏٙnʕ՟kӻ୕אል聯ӻ୕̙ٙቦ度f HLP:75 ɚධʱб̙ቦ度൙ПҦஔ 2 2003/05/25 ̙ቦ度Ҧஔ˓̅ ɚධʱб̙ቦ度൙ПҦஔ 3 ዚ率參數પ֛ ɚධ參數pʘᓃપ֛ ৿ண໊᜗̰ٙࣖ行މୌΥɚධʱбdՉ̰ࣖዚ率א不良率(˸ɨᔊ၈不良率)މpf若໊͟᜗פ՟ᅵ͉ɽʃމnٙפᅵ行މ݊ዹ立ٙd若ᅵ͉ٙᝈ಻ഐ؈̰ࣖ數މxdۆᅵx͉不良率މ໊᜗不良率ٙᓃપ֛࠽ˆpd͵уj nxˆ p=(5) 不良率ʘਜගપ֛ ͟ɚධʱбٙ累ጐʱбՌ數ක֐d xBin(x;n,p)=bin(k;n,p)∑k=0xn⎡⎤n−kk =p(1−p) (6) ∑⎢⎥kk=0⎣⎦xn!n−kk=p(1−p)∑k!(n−k)!k=0࿁pϾԊdBin(x;n,p)މ連ᚃՌ數dό(6)࿁pฆʱd੻Ցj xdBin(xnp)d⎡n⎤;.!n−kk=p(1−p)∑⎢⎥dpdpk!(n−k)!⎣k=0⎦xn!dn−kk=[p(1−p)]∑k!(n−k)!dpk=0 (7) xxn!n!n−k−1n−kkk−1=(n−k)p(1−p)(−1)+kp(1−p)∑∑k!(n−k)!k!(n−k)!k=0k=0xxn!n!n−k−1n−kkk−1=−p(1−p)+p(1−p)∑∑k!(n−k−1)!(k−1)!(n−k)!k=0k=1令ɪόʕഃ໮̛˙ୋɚ௅ʱٙk−1=rdۆ̙੻Ցj xx−1dBin(x;n,p)n!n!n−k−1n−r−1kr =−p(1−p)+p(1−p) (8) ∑∑dpk!(n−k−1)!r!(n−r−1)!k=0r=0ˢ༰ɪόഃ໮̛˙兩௅ʱ̙೯ତd兩٫̥݊Դ͜不Νٙਤᜊ數d兩٫ᅵόҁΌ޴Νdਬɓٙ不Νᓃ݊ୋɚ௅ʱˢୋɓ௅ʱˇɓධdऊ̘΍ஷධ̙੻Ցɨόj HLP:75 ɚධʱб̙ቦ度൙ПҦஔ 3 2003/05/25 ̙ቦ度Ҧஔ˓̅ ɚධʱб̙ቦ度൙ПҦஔ dBin(x;n,p)n!n−x−1x=−p(1−p)dpx!(n−x−1)!Γ(n+1)n−x−1x =−p(1−p) (9) Γ(x+1)Γ(n−x)1n−x−1x=−p(1−p)B(x+1,n−x)ɪόഃ໮̛˙ʕΓ(n+1)ձB(x+1,n−x)ʱйމТီՌ數ձԎ˼Ռ數dৰࠋ໮̮ٙ௅ʱމԎ˼ʱбٙዚ率੗度Ռ數f͵уj 1n−1m−1B(m,n)=t(1−t)dt∫0 (10) Γ(m)Γ(n)=Γ(m+n)∞m−1−x Γ(m)=xedx (11) ∫01n−1m−1 bet(x;m,n)=x(1−x) (12) B(m,n)ਗ਼ό(9)兩ᗙ࿁p੽0Їpᇍఖਂጐʱd̙੻Ցj ppdBin(x;n,z)1n−x−1x dz=−z(1−z)dz ∫∫00dzB(x+1,n−x)p1n−x−1x Bin(xn,p)−Bin(x;n,0)=−z(1−z)dz ∫0B(x+1,n−x)͟׵Bin(x;n,0)=1d˾ɝɪόd̙੻j p1n−x−1xBin(x;n,p)=1−z(1−z)dz∫0B(x+1,n−x)p=1−bet(z;x+1,n−x)dz∫ 0=1−Bet(p;x+1,n−x)1=bet(z;x+1,n−x)dz∫p令ν=2(x+1)eν=2(n−x)dF(ν,ν)d 12α212HLP:75 ɚධʱб̙ቦ度൙ПҦஔ 4 2003/05/25 ̙ቦ度Ҧஔ˓̅ ɚධʱб̙ቦ度൙ПҦஔ 2(x+1)Fα2;2(x+1),2(n−x)2(n−x)ˆp=L2(x+1)1+Fα2;2(x+1),2(n−x)2(n−x)Fα2;2(x+1),2(n−x)=2(n−x)+Fα2;2(x+1),2(n−x)2(x+1) 1=2(n−x)11+2(x+1)Fα2;2(x+1),2(n−x)1=2(n−x)1+F1−α2;2(n−x),2(x+1)2(x+1) ν2ˆ p= Lν+νF(ν,ν)21(1−α)12ν=2(n−r+1) 1ν=2r 2 4 ТီၾԎ˼Ռ數 ТီՌ數 HLP:75 ɚධʱб̙ቦ度൙ПҦஔ 5 2003/05/25 ̙ቦ度Ҧஔ˓̅ ɚධʱб̙ቦ度൙ПҦஔ ڝڌ1: ੬࿒ʱбʱЗ數ڌ ΦZ()ΦZ=PrZ≤z (){}Z0zZ Z HLP:75 ɚධʱб̙ቦ度൙ПҦஔ 1 2003/05/25 f(z) ̙ቦ度Ҧஔ˓̅ ɚධʱб̙ቦ度൙ПҦஔ ڝڌ2: tʱбʱЗ數ڌ Prt>t=α {}α2( )ʕ數࠽މఊᗙʱЗ數t α ν\α ν\α() () () () ()()()()()()() () ()1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 1011 1112 1213 1314 1415 1516 1617 1718 1819 1920 2021 2122 2223 2324 2425 2526 2627 2728 2829 2930 3040 4060 60120 ∞ ∞ HLP:75 ɚධʱб̙ቦ度൙ПҦஔ 2 2003/05/25 ̙ቦ度Ҧஔ˓̅ ɚධʱб̙ቦ度൙ПҦஔ ڝڌ3: ̔˙ʱбʱЗ數ڌ ν\α ν\α321 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 1011 1112 1213 1314 1415 1516 1617 1718 1819 1920 2021 2122 2223 2324 2425 2526 2627 2728 2829 2930 3040 4060 6080 HLP:75 ɚධʱб̙ቦ度൙ПҦஔ 3 2003/05/25 ̙ቦ度Ҧஔ˓̅ ɚධʱб̙ቦ度൙ПҦஔ ڝڌ4: ТီՌ數ڌ ∞x−1−tΓx=tedt ()∫0x Γx x Γx x Γx x Γx ()()()() Γx+1()ൗ1jΓx= ()xൗ2jΓx+1=xΓx ()()Γ()例1jΓ=== ()例2jΓ=×Γ=×= ()()例=××=××= ()()HLP:75 ɚධʱб̙ቦ度൙ПҦஔ 4 2003/05/25