2017-2018光滑學術大禮包by概率統(tǒng)計lecture1

1、Introduction to Probability and StatisticsInstructor: Minya Xu (徐敏亞)Phone: 62756274Email: Office: Room 365, Guanghua Building No. 2助教n楊光藝n張爽n楊冰Why leaning probability and statistics?nTED videonCase 1: 電視訪問一位能說會道的母親，她的孩子得了白電視訪問一位能說會道的母親，她的孩子得了白血病，他們住在高壓電線附近。血病，他們住在高壓電線附近。n住在高壓電線附近會導致兒童得白血病嗎？住在高壓電線附近會

2、導致兒童得白血病嗎？n在大眾心目中，往往相信軼聞。但我們應該要心存疑問n癌癥研究所花費5年的時間和500萬元為這個問題搜集資料。結論是：在白血病和暴露在高壓電線所產(chǎn)生的磁場之間，找不到相關關系。n數(shù)據(jù)比軼聞可靠，因為數(shù)據(jù)可以有系統(tǒng)地描繪出整體的情況，而軼聞只聚焦于少數(shù)特例。n好的數(shù)據(jù)勝過軼聞。比起軼聞和光大聲嚷嚷預測未來，數(shù)據(jù)要客觀得多。nCase 2: 對于便利連鎖店（如對于便利連鎖店（如7-11），店員的熱情招呼和），店員的熱情招呼和微笑服務會提高顧客滿意？微笑服務會提高顧客滿意？深入的數(shù)據(jù)分析證明這個想法完全錯誤的，彬彬有禮的店員反而降低了顧客滿意。可能的原因?nCase 3: 中國人工

3、作辛苦，工作時間長，工資收入低是有目共中國人工作辛苦，工作時間長，工資收入低是有目共睹的事實。中國人勤勞是全世界公認的美德。那么中國人的睹的事實。中國人勤勞是全世界公認的美德。那么中國人的勤勞能換來企業(yè)的高效率嗎？勤勞能換來企業(yè)的高效率嗎？行業(yè)世界500強排名公司名稱總部所在地營業(yè)額（百萬美元）員工數(shù)量人 均 營 業(yè)額 （ 百 萬美元）煉油1荷蘭皇家殼牌石油公司荷蘭484489.090,0005.382?？松梨诿绹?52926.099,1004.574英國石油公司英國386463.083,4004.635中國石油化工集團公司中國375214.01,021,9790.376中國石油天然氣集團公

4、司中國352338.01,668,0720.21 財富世界500強企業(yè) 將財富世界500強企業(yè)重新排序。排名首位的荷蘭，共有12家企業(yè)，平均人均營業(yè)額13.8百萬美元；中國進入世界500強企業(yè)共有79家，平均人均營業(yè)額0.6百萬美元，排名第二十六。差距為 23倍。 或許在過去，公司還能拍腦門做決策，無視正確的統(tǒng)計數(shù)據(jù)，因為他們還能靠跟隨行業(yè)“慣例”糊弄過去。如今，有效的利用統(tǒng)計數(shù)據(jù)是競爭的必要條件。更為重要的是，在你的競爭對手之前找到并充分利用統(tǒng)計數(shù)據(jù)，將成為打造你的競爭優(yōu)勢的關鍵所在。 如今的中國企業(yè)的管理正由粗放型轉向精細型管理。 統(tǒng)計分析能夠帶給你價值，對數(shù)據(jù)有敬畏之心的，率先進行投入的

5、企業(yè)，將更有機會在未來競爭中取得先機。 -哈佛商業(yè)評論nTextbooknProbability and Statistics, Revised Edition by Xiangzhong Fang, Ligang Lu, Dongfeng Li, Higher Education Press. nLecture Notesn （1） n （2）百度云盤：鏈接: mhSwGnI 密碼: t7i9nReferencesn 陳希孺，概率論與數(shù)理統(tǒng)計，中國科學技術大學出版社。課堂要求n不遲到，不早退不遲到，不早退n上課積極參與討論上課積極參與討論n積極關掉所有的電子產(chǎn)品積極關掉所有的電子產(chǎn)品n帶名牌

6、帶名牌Course AssessmentnYour final grade will be based on three components:nHomework assigned on a weekly basis will account for 30%. nYou may discuss homework problems with other students, but you must write them up independently. nHomework is due at the beginning of class on the due date. Note that h

7、omework that is turned in later than the end of class on the due date will not be graded. nIt 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 grade.nMidterm exam will accoun

8、t for 30%.nFinal exam will account for 40%.nNote that cheating is strictly not allowed and will result in zero score on the respective part.Introduction of ProbabilityExperiment(試驗)Suppose a coin is tossed once and the up face is recorded. The result we see and record is called an observation, or me

9、asurement, and the process of making an observation is called an experiment.The point is that a statistical experiment can be almost any act of observation as long as the e is uncertain. Definition: An experiment is an act or process of observation that leads to a single e that cannot be predicted w

10、ith certainty. 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 an

11、d only one of the possible experimental es will occur;there is uncertainty associated with which one will occur. ExperimentSample Point, Sample Space, EventnA basic possible e of an experiment is called a sample point (樣本點).nThe sample space (樣本空間) for an experiment is the set of all sample points.n

12、An event (事件) is a specific collection of sample points.nExperimentSample 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,

13、220The height is no less than 175cmnA given event is said to have occurred if the e of the experiment is one of the es in the event.nE.g., 拋骰子，S=1,2,3,4,5,6。n Event A: 拋出的是偶數(shù)，即 A=2,4,6。n 當觀察到s=2, 我們就說事件A發(fā)生了。Probability as Numerical Measure of UncertaintyProbability is a numerical measure of the like

14、lihood that an event will occur. Probabilities could be used as measures of the degree of uncertainty.概率n相對頻率（客觀概率）：n 多次重復做一個實驗時，那么其概率就是事件最終發(fā)生的次數(shù)比率。n個別概率（主觀概率）：n 生活中大部分事件不會重演，個別概率是個人對某種結果發(fā)生的主觀估計。 骰子可以一擲再擲The KP&L ProblemnKP&L company is starting a project designed to increase the generating

15、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 times f

16、or the design stage of 2, 3, or 4 months and completion times for the construction stage of 6, 7, or 8 months. nBecause 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 probability

17、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(

18、4,8)12ProbabilitiesnProbability can be interpreted as the relative frequency of the occurrence of an event if the experiment is repeated a large number of times.nThe KP&L problem:nCompletion Time (months) Number of Past Projects WithnStage 1 Stage 2 e Total Time These Completion Times Probabilit

19、yn2 6(2,6) 86 6/40= .15n2 7(2,7) 96 6/40= .15n2 8(2,8) 102 2/40= .05n3 6(3,6) 94 4/40= .10n3 7(3,7) 108 8/40= .20n3 8(3,8) 112 2/40= .05n4 6(4,6) 102 2/40= .05n4 7(4,7) 114 4/40= .10n4 8(4,8) 126 6/40= .15n T o t a l4 0Total 1.00nThe probability of the project being finished on time is n.15+.15+.05+

20、.10+.20+.05=.701.4 Set TheoryProbability TheorySet Theorysample space 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

21、possible es in SA=100,101,102,Relations of Set TheorynAn e s is a member of SnEvent A is contained in event B: every e that belongs to the subset defining A also belongs to the subset defining BnEmpty set: the subset that contains no esSs BA BAAB,BA CACB,BA SA Operations of Set TheorynUnions. The un

22、ion of A and B is defined to be the event containing all es that belong to either A or B, or the event that either A or B would occur.n e.g.1,2,.,100,51,52,.,1501,2,.,150ABABBBABASSA,AAAAA,ABBA )()(CBACBACBA the union of n events is defined to be the event which contains all es that belong to at lea

23、st 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.niniAAAA211211AAAiiSome NotationnIntersections. The intersection of A and B is defined to be the event containing all es that belong both to A and B, or the event that

24、both A and B would occur.ABABAASA,AAAA,ABBA )()()()()()()()(CABACBACABACBACBACBACBAe.g.1,2,.,100,51,52,.,15051,52,.,100ABAB 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 occur. the inter

25、section of an infinite sequence of events.niniAAAA211211AAAiiSome NotationnComplements. 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.ne.g.1,3,56 , 5 , 4 , 3 , 2 , 1,6 , 4 , 2cASA ccccccAA,SAASS,A)A(nDisjoint

26、events（不相交事件，互斥事件）. A and B are disjoint, or mutually exclusive, if they have no es in common. nMore generally, a collection of events A1, A2, . . . , An is said to be disjoint/mutually exclusive if no two of them have any es in common.n n BA1.5 Definition of ProbabilitynDefinition: A probability me

27、asure, or simply a probability, on a sample space S is a specification of numbers Pr(A) for all events A that satisfy Axioms 1, 2, and 3.Commentn1933年，蘇聯(lián)大數(shù)學家Kolmogorov制定了這個概率論的公理體系，其中對概率是什么不加定義，只指出關于其運算所必須遵守的幾條規(guī)則，這樣就回避了如何定義概率這個難題，在這個基礎上，建立起了概率論的宏偉大廈。n現(xiàn)今談概率，不論是客觀主觀，大都遵守柯氏的公理體系，這有極大的好處：不論對概率的本質理解有何不同，

28、在運算推理上大家都遵守公認的準則，而不是各行其是。nTheorem 1.5.1n 0)Pr( Theorem 1.5.2 For any finite sequence of disjoint events A1, ., An, niiiniAA11)Pr()Pr(nTheorem 1.5.3 For any event A,)Pr(1)Pr(AAcnTheorem 1.5.4 For any event A,1)Pr(0AnTheorem 1.5.5 If , thenn BA)Pr()Pr(BA nTheorem 1.5.6 For any two events A and B,nn)Pr()Pr()Pr()Pr(ABBABA=+Example: Diagnosing DiseasesnA patient arrives