No14概率統(tǒng)計(jì)模型

1、概率統(tǒng)計(jì)模型 幾個(gè)概率分布 幾個(gè)數(shù)學(xué)模型 均勻分布1axbf ( x )ba0 其其他他（一維連續(xù)型）（一維連續(xù)型）（二維連續(xù)型）（二維連續(xù)型） kx, yDf ( x, y )0 其其他他（多維連續(xù)型）（多維連續(xù)型）kxDf ( x )0 其其他他 T12nxx ,x ,x 使用場合：使用場合：1）該問題具有某種均勻性；）該問題具有某種均勻性；2）該問題無任何先驗(yàn)信息。）該問題無任何先驗(yàn)信息。 二項(xiàng)分布二項(xiàng)分布 設(shè)試驗(yàn)設(shè)試驗(yàn) E 只有兩個(gè)結(jié)果：只有兩個(gè)結(jié)果：和和 ，記記: :獨(dú)立地重復(fù)獨(dú)立地重復(fù) n 次。次。P( A)p,P( A)1pq(0p1) Akkn knP(Xk)C p (1p)k

2、1,2,n L LX B( n, p )記作：記作： 只有兩個(gè)結(jié)果：只有兩個(gè)結(jié)果：和和 ，A使用場合：使用場合：獨(dú)立地重復(fù)獨(dú)立地重復(fù) n 次。次。 Poission 分布分布kttk1,P2, Xk ek! L LX( t ) : :使用場合：使用場合：具有：具有： 平穩(wěn)性 獨(dú)立增量性 普通性22( x)1f ( x )ex2 正態(tài)分布（正態(tài)分布（normal）2X N(,) 1222( x)12ye y三倍標(biāo)準(zhǔn)差原則：P X3 99.74%可以認(rèn)為，可以認(rèn)為，Y 的取值幾乎全部集中在的取值幾乎全部集中在3 ,3 區(qū)間內(nèi)。區(qū)間內(nèi)。使用場合：使用場合：滿足中心極限定理?xiàng)l件。滿足中心極限定理?xiàng)l件。

3、12122122122122x11f ( x,y )exp ()2(1)21xyy2 ()()() 其中其中均為常數(shù)均為常數(shù),且且1212,120,0, | 1 二維正態(tài)分布12122122122122x11f ( x,y )exp ()2(1)21xyy2 ()()() 2112TT122212xx, y, 1212cov X ,Y 2121cov Y ,X T12 21 211fx,yexp( x)( x)(2)|2 11121n21222nn1n2nn 1122nnxxx,x iii jijEX ,cov X , X T1n 21 211fx, yexp( x)( x)(2)|2 Air

4、line Overbooking（2002 MCM)Youre all packed and ready to go on a trip to visit your best friend in New York City. After you check in at the ticket counter, the airline clerk announces that your fright has been overbooked passengers need to check in immediately to determine if they still have a seat.H

5、istorically, airlines know that only a certain percentage of passengers who have made reservations on a particular flight will actually take that flight. Consequently, most airlines overbook that is, they take more reservations than the capacity of the aircraft. Occasionally, more passengers will wa

6、nt to take a flight than the capacity of the plane, leading to one or more passengers being bumped and thus unable to take the flight for which they had reservations.Airlines deal with bumped passengers in various ways. Some are given nothing, some are booked on later flights on other airlines, and

7、some are given some kind of cash or airline ticket incentive.Consider the overbooking issue in light of the current situation: fewer flights by airlines from point A to point B; heightened security at and around airports, passengers fear, and loss of billions of dollars in revenue by airlines to dat

8、e.Build a mathematical model that examines the effects that different overbooking schemes have on the revenue received by an airline company, in order to find an optimal overbooking strategy -that is, the number of people by which an airline should overbook a particular flight so that the companys r

9、evenue is maximized. Ensure that your model reflects the issues above and consider alternatives for handling bumped passengers. Additionally, write a short memorandum to the airlines CEO summarizing your findings and analysis.Technical Considerations If the airline arranges substitute transportation

10、 such that the passenger will reach his/her destination within one hour of the original flights arrival time, there is no obligatory compensation. If the airline arranges substitute transportation such that the passenger will reach his/her destination between one and two hours after the original fli

11、ghts arrival time, the airline must pay the passenger an amount equal to the one-way fare for flight to the final destination. Frontier offers only one class of service to all passengers. Thus, we base our overbooking models on single-class aircraft. If the substitute transportation is scheduled to

12、arrive any later than two hours after the original flights arrival time, or if the airline does not make any substitute travel arrangements, the airline must pay an amount equal to twice the cost of the fare to the final destination. We need concern ourselves only with the sale of restricted tickets

13、. Frontiers are nonrefundable, save for the ability to transfer to another Frontier flight for $60 Frontier 2001. Restricted tickets represent all but a very small percentage of all tickets, and many ticket brokers, such as P, sell only restricted tickets. Airline booking considerations are frequent

14、ly based on the break-even load-factor, a percentage of airplane seat capacity that must be filled to acquire neither loss or profit on a particular flight. The break-even load-factor for Flight 502 in 2001 was 57.8%.iiB iBPr passenger arrive at the gate Pr Xi C p (1p)i pflight$Add$Add$T ( X )(BX )R

15、+AirfareXCostXC(AirfareCost) ( XC )CXC(AirfareCost) ( XC )Bump( XC )XC flight$AddR=$60; Airfare=$316; Cost$24,648; C134; C13457.8%78; Cost$16 i1,2,Bp0.88 No Overbooking Bump Threshold Model Linear Compensation PlanPr XC BT(0.05 ) $Bump( XC )B( XC )Modeling: Nonlinear Compensation Planr (X C)$Bump( X

16、C )B ( XC )e Time-Dependent Compensation Plane0.07324t3160t15minCompensation( t )=105.33e15mint30min X Cii=1Bump( XC )Compensation( t ) Results: No OverbookingpBpC.88134118 (134118 ) $60(11878 ) $300$12,960 Lost opportunity cost $300*16 = $4,800 ! Bump Threshold ModelPr XC BT(0.05 ) B145 Linear Comp

17、ensation Plan$Bump( XC )B( XC )0.042( X C )316e( XC ) 0.134( X C )50e( XC ) 0.100( X C )100e( XC ) 0.065( X C )200e( XC ) Nonlinear Compensation Planr (X C)$Bump( XC )B ( XC )e Time-Dependent Compensation PlaneProfitability is maximized around B=160一件產(chǎn)品由若干零件組裝而成，標(biāo)志產(chǎn)品性能的某個(gè)參數(shù)一件產(chǎn)品由若干零件組裝而成，標(biāo)志產(chǎn)品性能的某個(gè)參數(shù)取

18、決于這些零件的參數(shù)，零件參數(shù)包括標(biāo)定值和容差兩部取決于這些零件的參數(shù)，零件參數(shù)包括標(biāo)定值和容差兩部分進(jìn)行成批生產(chǎn)時(shí)，標(biāo)定值表示一批零件該參數(shù)的平均值，分進(jìn)行成批生產(chǎn)時(shí)，標(biāo)定值表示一批零件該參數(shù)的平均值，容差則給出了參數(shù)偏離其標(biāo)定值的容許范圍若將零件參數(shù)容差則給出了參數(shù)偏離其標(biāo)定值的容許范圍若將零件參數(shù)視為隨機(jī)變量，則標(biāo)定值代表期望值，在生產(chǎn)部門無特殊要視為隨機(jī)變量，則標(biāo)定值代表期望值，在生產(chǎn)部門無特殊要求時(shí)，容差通常規(guī)定為均方差的求時(shí)，容差通常規(guī)定為均方差的3倍。倍。進(jìn)行零件參數(shù)設(shè)計(jì)，就是要確定其標(biāo)定值和容差這時(shí)要進(jìn)行零件參數(shù)設(shè)計(jì)，就是要確定其標(biāo)定值和容差這時(shí)要考慮兩方面因素；考慮兩方面因素；

19、零件的參數(shù)設(shè)計(jì)零件的參數(shù)設(shè)計(jì)(CUMCM 1997年年A題題)一、當(dāng)各零件組裝成產(chǎn)品時(shí)，如果產(chǎn)品參數(shù)偏離預(yù)先設(shè)定的一、當(dāng)各零件組裝成產(chǎn)品時(shí)，如果產(chǎn)品參數(shù)偏離預(yù)先設(shè)定的目標(biāo)值，就會(huì)造成質(zhì)量損失，偏離越大，損失越大；目標(biāo)值，就會(huì)造成質(zhì)量損失，偏離越大，損失越大；二、零件容差的大小決定了其制造成本，容差設(shè)計(jì)得越小，二、零件容差的大小決定了其制造成本，容差設(shè)計(jì)得越小，成本越高。成本越高。試通過如下的具體問題給出一般的零件參數(shù)設(shè)計(jì)方法：試通過如下的具體問題給出一般的零件參數(shù)設(shè)計(jì)方法：粒子分離器某參數(shù)粒子分離器某參數(shù)( y )由由7個(gè)零件的參數(shù)個(gè)零件的參數(shù)( x1，x2，x7 )決決定，經(jīng)驗(yàn)公式為定，經(jīng)驗(yàn)

20、公式為:3 / 2440.85223132167xx12.62 10.36xxxxy174.42xxxx x y 的目標(biāo)值的目標(biāo)值( y0 )為為1.50當(dāng)當(dāng) y 偏離偏離y00.1時(shí)，產(chǎn)品為時(shí)，產(chǎn)品為次品，質(zhì)量損失為次品，質(zhì)量損失為1000元；當(dāng)元；當(dāng)y偏離偏離y00.3時(shí)，產(chǎn)品為廢品，時(shí)，產(chǎn)品為廢品，損失為損失為9000元元A等為等為l，B等為等為5，C等為等為l0， 現(xiàn)進(jìn)行成批生產(chǎn)，每批產(chǎn)量現(xiàn)進(jìn)行成批生產(chǎn)，每批產(chǎn)量1000個(gè)，在原設(shè)計(jì)中，個(gè)，在原設(shè)計(jì)中，7個(gè)零件參數(shù)的標(biāo)定值為：個(gè)零件參數(shù)的標(biāo)定值為：x1=0.1, x2=0.3,x3=0.1 x4=0.1, x5=1.5 x6=16, x

21、7=0.75;容差均取最便宜的等級(jí)。容差均取最便宜的等級(jí)。 請(qǐng)你綜合考慮請(qǐng)你綜合考慮 y 偏離偏離 y0 造成的損失和零件成造成的損失和零件成本，重新設(shè)計(jì)零件參數(shù)本，重新設(shè)計(jì)零件參數(shù)(包括標(biāo)定值和容差包括標(biāo)定值和容差)，并，并與原設(shè)計(jì)比較，總費(fèi)用降低了多少。與原設(shè)計(jì)比較，總費(fèi)用降低了多少。損失函數(shù)損失函數(shù) 10000yy0.1L( y )10000.1yy0.39000yy0.3 損失函數(shù)損失函數(shù) 2250L( y )k( yy )k10 基本模型基本模型0Z( x ,t )E L( y ) + C ( t ) 7iii 1C(t )ct 070iii0ii 1ixfyf( x )d ( xx

22、 )dx 520L( y )10 ( yy ) 75200iii0i 17252200iii0i 1E L( y ) 10 E f ( x )yd ( xx )10f ( x )yd E( xx ) 222ii0ii0it xE( x -x )3 7725222000ii0 iiii 1i 11Z x ,t10f ( x )yd x tct9 在在中，求 使：*0 x ,t *00Z x , tminZ x ,t 自動(dòng)化車床管理自動(dòng)化車床管理CUMCMl999年年A題題) 一道工序用自動(dòng)化車床連續(xù)加工某種零件，由于刀一道工序用自動(dòng)化車床連續(xù)加工某種零件，由于刀具損壞等原因該工序會(huì)出現(xiàn)故障，其中

23、刀具損壞故障占具損壞等原因該工序會(huì)出現(xiàn)故障，其中刀具損壞故障占95，其他故障僅占，其他故障僅占5工序出現(xiàn)故障是完全隨機(jī)的，工序出現(xiàn)故障是完全隨機(jī)的，假定在生產(chǎn)任一零件時(shí)出現(xiàn)故障的機(jī)會(huì)均相同工作人假定在生產(chǎn)任一零件時(shí)出現(xiàn)故障的機(jī)會(huì)均相同工作人員通過檢查零件來確定工序是否出現(xiàn)故障員通過檢查零件來確定工序是否出現(xiàn)故障 現(xiàn)積累有現(xiàn)積累有100次刀具故障記錄，故障出現(xiàn)時(shí)該刀具完次刀具故障記錄，故障出現(xiàn)時(shí)該刀具完成的零件數(shù)如附表現(xiàn)計(jì)劃在刀具加工一定件數(shù)后定期成的零件數(shù)如附表現(xiàn)計(jì)劃在刀具加工一定件數(shù)后定期更換新刀具更換新刀具 已知生產(chǎn)工序的費(fèi)用參數(shù)如下：已知生產(chǎn)工序的費(fèi)用參數(shù)如下： 故障時(shí)產(chǎn)出的零件損失費(fèi)用

24、故障時(shí)產(chǎn)出的零件損失費(fèi)用 f=200 元元/件件 進(jìn)行檢查的費(fèi)用進(jìn)行檢查的費(fèi)用 t=10 元次；元次； 發(fā)現(xiàn)故障進(jìn)行調(diào)節(jié)使恢復(fù)正常的平均費(fèi)用發(fā)現(xiàn)故障進(jìn)行調(diào)節(jié)使恢復(fù)正常的平均費(fèi)用d=3000 元元/次次 (包括刀具費(fèi)包括刀具費(fèi))； 未發(fā)現(xiàn)故障時(shí)更換一把新刀具的費(fèi)用未發(fā)現(xiàn)故障時(shí)更換一把新刀具的費(fèi)用 k=1000 元元/次次 假定工序故障時(shí)產(chǎn)出的零件均為不合格品，正常時(shí)產(chǎn)假定工序故障時(shí)產(chǎn)出的零件均為不合格品，正常時(shí)產(chǎn)出的零件均為合格品。試對(duì)該工序設(shè)計(jì)效益最好的檢出的零件均為合格品。試對(duì)該工序設(shè)計(jì)效益最好的檢查間隔查間隔(生產(chǎn)多少零件檢查一次生產(chǎn)多少零件檢查一次)和刀具更換策略。和刀具更換策略。 如果

25、該工序正常時(shí)產(chǎn)出的零件不全是合格品，有如果該工序正常時(shí)產(chǎn)出的零件不全是合格品，有2為不合格品；而工序故障時(shí)產(chǎn)出的零件有為不合格品；而工序故障時(shí)產(chǎn)出的零件有40為合格為合格品，品，60%為不合格品工序正常而誤認(rèn)有故障停機(jī)產(chǎn)為不合格品工序正常而誤認(rèn)有故障停機(jī)產(chǎn)生的損失費(fèi)用為生的損失費(fèi)用為1500元次對(duì)該工序設(shè)計(jì)效益最好元次對(duì)該工序設(shè)計(jì)效益最好的檢查間隔和刀具更換策略。的檢查間隔和刀具更換策略。 在在2. 的情況可否改進(jìn)檢查方式獲得更高的效益。的情況可否改進(jìn)檢查方式獲得更高的效益。附：附：100次刀具故障記錄次刀具故障記錄(完成的零件數(shù)完成的零件數(shù))零件損失費(fèi)用：零件損失費(fèi)用： f檢查的費(fèi)用：檢查的費(fèi)用： t調(diào)節(jié)平均費(fèi)用：調(diào)節(jié)平均費(fèi)用：d更換新刀具費(fèi)用更換新刀具費(fèi)用 ：k檢查間隔：檢