經(jīng)管學(xué)會(huì)網(wǎng)盤光華方向?qū)I(yè)概統(tǒng)lecture

1、Introduction to Probability and StatisticsInstructor: Minya Xu (徐敏亞)Phone: 62756274Email: Office: Room 365, Guanghua Building No. 2“I keep saying that the sexy job in the next 10 years will be statisticians, and Im not kidding.” Hal Varian, chief economist at Google. Application fields of probabilit

2、y and statistics:EconomicsFinanceMarketingAccountingOrganizational BehaviorMedicineGeneticsPhysicsChemistryMeteorologyPhotography from satellitesProbability theory and statistics provide important methodology in most areas of engineering, science, and management.TextbookProbability and Statistics, R

3、evised Edition by Xiangzhong Fang, Ligang Lu, Dongfeng Li, Higher Education Press, 2005. Lecture NotesReferencesMorris H. DeGroot and Mark Schervish (2001), Probability and Statistics, Addison Wesley, 3rd edition.John A. Rice (1994), Mathematical Statistics and Data Analysis, 2nd edition, Duxbury Pr

4、ess.陳希孺(2000) ，概率論與數(shù)理統(tǒng)計(jì)，中國(guó)科學(xué)技術(shù)大學(xué)出版社。Course AssessmentYour final grade will be based on three components:Homework assigned on a weekly basis will account for 30%. You may discuss homework problems with other students, but you must write them up independently. Homework is due at the beginning of class

5、 on the due date. Note that homework that is turned in later than the end of class on the due date will not be graded. It is understood that from time to time your schedule may not allow you to turn in your homework on time, so your lowest homework score will be dropped when computing your final gra

6、de.Midterm exam will account for 30%.Final exam will account for 40%.Note that cheating is strictly not allowed and will result in zero score on the respective part.Experiment(試驗(yàn))Definition: An experiment is an act or process of observation that leads to a single e that cannot be predicted with cert

7、ainty. e.g., Consider the experiment of counting the number of customers at a restaurant on a particular day. The basic possible es of this experiment are 0,1, 2, 3, ExperimentThe features of an experiment are:each of its possible es can be specified before the experiment is performed;one and only o

8、ne of the possible experimental es will occur;there is uncertainty associated with which one will occur. ExperimentSample Point, Sample Space, EventA basic possible e of an experiment is called a sample point (樣本點(diǎn)).The sample space (樣本空間) for an experiment is the set of all sample points.An event (事

9、件) is a specific collection of sample points.ExperimentSample SpaceEventThe number of customers at a restaurant on a particular day0,1,2,The number of customers is at least 100Select a part for inspectionDefective, NondefectiveDefectiveThe height of a randomly chosen PKU student (cm)120,220The heigh

10、t is no less than 175cmProbability as Numerical Measure of UncertaintyProbability is a numerical measure of the likelihood that an event will occur. Probabilities could be used as measures of the degree of uncertainty.The KP&L ProblemKP&L company is starting a project designed to increase the genera

11、ting capacity of one of its plants. The project is divided into two sequential stages: stage 1 (design) and stage 2 (construction). Management cannot predict beforehand the exact time required to complete each stage of the project. An analysis of similar construction projects has shown completion ti

12、mes for the design stage of 2, 3, or 4 months and completion times for the construction stage of 6, 7, or 8 months. Because of the critical need for additional electrical power, management has set a goal of 10 months for the completion of the entire project, and management wants to know the probabil

13、ity that the project will be finished on time.Sample Space for the KP&L ProblemThere are (3)(3)=9 es in the sample space.Completion Time (months)Stage 1 Stage 2 Total Project(Design) (Construction) Experimental e Completion Time26(2,6)827(2,7)928(2,8)1036(3,6)937(3,7)1038(3,8)1146(4,6)1047(4,7)1148(

14、4,8)12ProbabilitiesProbability can be interpreted as the relative frequency of the occurrence of an event if the experiment is repeated a large number of times.The KP&L problem:Completion Time (months) Number of Past Projects WithStage 1 Stage 2 e Total Time These Completion Times Probability2 6(2,6

15、) 86 6/40= .152 7(2,7) 96 6/40= .152 8(2,8) 102 2/40= .053 6(3,6) 94 4/40= .103 7(3,7) 108 8/40= .203 8(3,8) 112 2/40= .054 6(4,6) 102 2/40= .054 7(4,7) 114 4/40= .104 8(4,8) 126 6/40= .15 Total40Total 1.00The probability of the project being finished on time is .15+.15+.05+.10+.20+.05=.70Mathematic

16、al Theory of ProbabilityPrecondition: probabilities are assigned to each possible e in an experiment.Two problems:Determining the probabilities of certain events.Revising the probabilities of events when additional relevant information is obtained.1.4 Set TheoryProbability TheorySet Theorysample spa

17、ce of an experimente.g. The number of customers at a restaurant on a particular daythe set of all possible es SS=0,1,2,3,an e (a sample point)e.g. 100 customersan element in the set S100an evente.g. at least 100 customersa subset of possible es in SA=100,101,102,Relations of Set TheoryAn e s is a me

18、mber of SEvent A is contained in event B: every e that belongs to the subset defining A also belongs to the subset defining BEmpty set: the subset that contains no esOperations of Set TheoryUnions. The union of A and B is defined to be the event containing all es that belong to either A or B, or the

19、 event that either A or B would occur. the union of n events is defined to be the event which contains all es that belong to at least one of these n events, or the event that at least one of these n events would occur. the union of an infinite sequence of events.Some NotationIntersections. The inter

20、section of A and B is defined to be the event containing all es that belong both to A and B, or the event that both A and B would occur. the intersection of n events is defined to be the event which contains all es that are common to all these n events, or the event that all these n events would occ

21、ur. the intersection of an infinite sequence of events.Some NotationComplements. The complement of A is defined to be the event containing all es in S that do not belong to A, or the event that event A would not occur.e.g.Disjoint events（不相交事件，互斥事件）. A and B are disjoint, or mutually exclusive, if t

22、hey have no es in common. e.g. partition of the sample space into eight disjoint events according to three events A1, A2, A3Axiom 1. For any event A, Axiom 2.Axiom 3. For every infinite sequence of disjoint events A1, A2, . 1.5 Definition of ProbabilityNeed to assign to each event A a number Pr(A) indicating the probability of A.A probability distribution on a sample space S is a specification of numbers Pr(A) which satisfy Axioms 1, 2, and 3.Probability DistributionTheorem 1.5.1Theorem 1.5.2 For any finite sequence of di