概率论与数理统计英文版总结.docx

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1、精品名师归纳总结Sample Space 样本空间The set of all possible outcomes of a statistical experiment is called thesample space.Event 大事An eventis a subset of a sample space.certain event（必定大事）：The sample spaceS itself, is certainly an event, which is called a certain event, means that it always occurs in the exper

2、iment.impossible event（不行能大事）：The empty set,denotedby, isalsoan event,calledan impossibleevent,means that it never occurs in the experiment.Probability of events（概率）If the number of successes inn trails is denoted bys , and if the sequence of可编辑资料 - - - 欢迎下载精品名师归纳总结relativefrequenciess/ n obtainedfo

3、rlargerand largervalueofn approachesa可编辑资料 - - - 欢迎下载精品名师归纳总结limit, then this limit is defined as theprobabilityof success in a single trial.“ equally likely to occur”-probability（古典概率）Ifa sample spaceS consistsofN samplepoints,each isequallylikelytooccur. Assume that the eventA consists ofn sample

4、points, then theprobabilityp可编辑资料 - - - 欢迎下载精品名师归纳总结that A occurs ispP An N可编辑资料 - - - 欢迎下载精品名师归纳总结Mutually exclusive互斥大事 Definition2.4 . 1EventsA1, A2, Anarecalledmutuallyexclusive,if可编辑资料 - - - 欢迎下载精品名师归纳总结AiAj,ij .Theorem 2.4.1IfA andB are mutually exclusive, then可编辑资料 - - - 欢迎下载精品名师归纳总结P ABP APB

5、 2.4.1可编辑资料 - - - 欢迎下载精品名师归纳总结Mutually independent大事的独立性Two eventsA andB are said to beindependentif可编辑资料 - - - 欢迎下载精品名师归纳总结P ABP APB可编辑资料 - - - 欢迎下载精品名师归纳总结Or Two eventsA andB are independent if and only ifP B | APB .可编辑资料 - - - 欢迎下载精品名师归纳总结可编辑资料 - - - 欢迎下载精品名师归纳总结Conditional Probability条件概率The pro

6、bability of an event is frequently influenced by other events.DefinitionThe conditionalprobabilityofB , givenA , denoted byP B | A , is可编辑资料 - - - 欢迎下载精品名师归纳总结可编辑资料 - - - 欢迎下载精品名师归纳总结defined byPB | AP ABifP AP A0 .2.5.1可编辑资料 - - - 欢迎下载精品名师归纳总结The multiplication theorem乘法定理可编辑资料 - - - 欢迎下载精品名师归纳总结IfA

7、1,A2,Akare events, then可编辑资料 - - - 欢迎下载精品名师归纳总结可编辑资料 - - - 欢迎下载精品名师归纳总结PA1A2Ak PA1PA2|A1P A3| A1A2P Ak |A1A2Ak 1可编辑资料 - - - 欢迎下载精品名师归纳总结可编辑资料 - - - 欢迎下载精品名师归纳总结IftheeventsA1 ,A2,Akareindependent,thenforanysubset可编辑资料 - - - 欢迎下载精品名师归纳总结 i1,i2 ,im 1, 2, k ,PAAAPA PA P Ai 1i 2i mi 1i 2i m可编辑资料 - - - 欢迎

8、下载精品名师归纳总结 全概率公式 total probability Theorem 2.6.1.If the eventsB1, B2 , Bkconstitute a partition of the可编辑资料 - - - 欢迎下载精品名师归纳总结可编辑资料 - - - 欢迎下载精品名师归纳总结sample space S such thatofS ,P B j 0forj1,2, k,thanforany eventA可编辑资料 - - - 欢迎下载精品名师归纳总结可编辑资料 - - - 欢迎下载精品名师归纳总结P AkkPABj PB j P ABj 可编辑资料 - - - 欢迎下载精

9、品名师归纳总结2.6.2j 1j 1可编辑资料 - - - 欢迎下载精品名师归纳总结（贝叶斯公式 Bayesformula.）可编辑资料 - - - 欢迎下载精品名师归纳总结Theorem 2.6.2If the eventsB1, B2, Bkconstitute a partition of the可编辑资料 - - - 欢迎下载精品名师归纳总结可编辑资料 - - - 欢迎下载精品名师归纳总结sample space S such thatPB j 0forj1,2, k,than for any可编辑资料 - - - 欢迎下载精品名师归纳总结event A of S,P A0 ,可编辑资

10、料 - - - 欢迎下载精品名师归纳总结P Bi| AP Bi P A | Bi kP Bj P A | B j .fori1,2, k可编辑资料 - - - 欢迎下载精品名师归纳总结j 12.6.2ProofBy the definition of conditional probability,可编辑资料 - - - 欢迎下载精品名师归纳总结可编辑资料 - - - 欢迎下载精品名师归纳总结PBi| APBiAP A可编辑资料 - - - 欢迎下载精品名师归纳总结Using the theorem of total probability, we have可编辑资料 - - - 欢迎下载精品

11、名师归纳总结PBi| APBi P A | Bi kP Bj P A | B j i1,2, k可编辑资料 - - - 欢迎下载精品名师归纳总结j 11. random variable definitionDefinition 3.1.1A random variableis a real valued function definedon a sample space; i.e.itassignsa realnumber to each sample point in the sample space.2. Distribution functionDefinition 3.1.2LetX

12、 be a random variable on the sample space S . Then the function可编辑资料 - - - 欢迎下载精品名师归纳总结F X P Xx .xR可编辑资料 - - - 欢迎下载精品名师归纳总结is called thedistribution functionofX可编辑资料 - - - 欢迎下载精品名师归纳总结Note The distributionfunctionF X isdefinedon realnumbers,可编辑资料 - - - 欢迎下载精品名师归纳总结not on sample space.3. Properties可编

13、辑资料 - - - 欢迎下载精品名师归纳总结The distributionfunctionF xofa random variableXhas可编辑资料 - - - 欢迎下载精品名师归纳总结the following properties:可编辑资料 - - - 欢迎下载精品名师归纳总结1F xis non-decreasing.可编辑资料 - - - 欢迎下载精品名师归纳总结可编辑资料 - - - 欢迎下载精品名师归纳总结In fact, ifx1x2 , then the event Xx1is a subset of可编辑资料 - - - 欢迎下载精品名师归纳总结可编辑资料 - - -

14、 欢迎下载精品名师归纳总结the event Xx2,thus可编辑资料 - - - 欢迎下载精品名师归纳总结可编辑资料 - - - 欢迎下载精品名师归纳总结F x1P Xx1P Xx2 F x2 可编辑资料 - - - 欢迎下载精品名师归纳总结可编辑资料 - - - 欢迎下载精品名师归纳总结2F limxF limxF x0 ,F x1 .可编辑资料 - - - 欢迎下载精品名师归纳总结可编辑资料 - - - 欢迎下载精品名师归纳总结3For anyx0R ,limxx0F x0F x00F x0 .This is to say,可编辑资料 - - - 欢迎下载精品名师归纳总结可编辑资料 -

15、 - - 欢迎下载精品名师归纳总结thedistributionfunctionF xofa random variableXis可编辑资料 - - - 欢迎下载精品名师归纳总结right continuous.3.2 Discrete Random Variables离散型随机变量Definition 3.2.1A random variableXis called adiscrete random variable, ifittakes values from a finiteset or,a set可编辑资料 - - - 欢迎下载精品名师归纳总结whose elements can be

16、 written as a sequence a1, a2,an,可编辑资料 - - - 欢迎下载精品名师归纳总结geometric distribution（几何分布） X 1 2 3 4 k P p q1p q2p q3 p qk 1p Binomial distribution（二项分布）Definition 3.4.1The numberXof successes inn Bernoulli trials is called abinomial random variable. The probability distribution of this discrete random v

17、ariable is called thebinomial distributionwith parametersn andp , denoted byB n, p .poisson distribution（泊松分布）Definition 3.5.1A discrete random variableXis called a可编辑资料 - - - 欢迎下载精品名师归纳总结Poissonrandom variable,ifittakesvaluesfromtheset0,1, 2, , and if可编辑资料 - - - 欢迎下载精品名师归纳总结P Xk kpk;e,0k .k0,1,2,可编

18、辑资料 - - - 欢迎下载精品名师归纳总结3.5.1Distribution 3.5.1 is called thePoisson distributionwith parameter, denoted byP .Expectation mean数学期望Definition 3.3.1LetXbe a discrete random variable. The expectationormean ofX is defined as可编辑资料 - - - 欢迎下载精品名师归纳总结E X xP Xxx3.3.1可编辑资料 - - - 欢迎下载精品名师归纳总结2. Variance方差standa

19、rd deviation（标准差）Definition3.3.2LetXbe a discreterandom variable,having可编辑资料 - - - 欢迎下载精品名师归纳总结expectationE X . Then the varianceofX , denote byD X 可编辑资料 - - - 欢迎下载精品名师归纳总结可编辑资料 - - - 欢迎下载精品名师归纳总结is defined as the expectation of the random variable X2可编辑资料 - - - 欢迎下载精品名师归纳总结可编辑资料 - - - 欢迎下载精品名师归纳总结D

20、 X E X 23.3.6可编辑资料 - - - 欢迎下载精品名师归纳总结可编辑资料 - - - 欢迎下载精品名师归纳总结The square rootof the varianceD X , denote byD X ,可编辑资料 - - - 欢迎下载精品名师归纳总结可编辑资料 - - - 欢迎下载精品名师归纳总结iscalledthestandarddeviationofX :122D X E X可编辑资料 - - - 欢迎下载精品名师归纳总结3.3.7probability density function概率密度函数Definition 4.1.1A functionf x defin

21、ed on, is called aprobability可编辑资料 - - - 欢迎下载精品名师归纳总结density function（概率密度函数 ） if:可编辑资料 - - - 欢迎下载精品名师归纳总结if x0 for anyxR ;可编辑资料 - - - 欢迎下载精品名师归纳总结iif x is intergrable 可积的 on, andf xdx1 .Definition 4.1.2Letf x be a probability density function. IfX is a random variable having distribution function可编

22、辑资料 - - - 欢迎下载精品名师归纳总结F xP Xxxf t dt ,4.1.1可编辑资料 - - - 欢迎下载精品名师归纳总结thenX is called acontinuous random variablehaving density functionf x. In this case,可编辑资料 - - - 欢迎下载精品名师归纳总结P x1Xx2 x2f t dt .4.1.2x1可编辑资料 - - - 欢迎下载精品名师归纳总结5. Mean （均值）Definition 4.1.2 Let X be a continuous random variable having pr

23、obability density function fx. Then the mean or expectation of X is defined byE X xf xdx ,4.1.36. variance方差Similarly, the variance and standard deviation of a continuous random variableXis defined by可编辑资料 - - - 欢迎下载精品名师归纳总结2D X E X 2 ,4.1.4可编辑资料 - - - 欢迎下载精品名师归纳总结可编辑资料 - - - 欢迎下载精品名师归纳总结WhereE X is

24、 the mean ofX,is referred to as thestandard deviation.可编辑资料 - - - 欢迎下载精品名师归纳总结We easily get可编辑资料 - - - 欢迎下载精品名师归纳总结2D X x2 fxdx2 . 4.1.5可编辑资料 - - - 欢迎下载精品名师归纳总结.4.2 Uniform Distribution匀称分布The uniformdistribution,withthe parametersa and b,has probabilitydensity function可编辑资料 - - - 欢迎下载精品名师归纳总结f x1fo

25、rbaaxb,可编辑资料 - - - 欢迎下载精品名师归纳总结0elsewhere,4.5 Exponential Distribution指数分布Definition4.5.1 A continuous variable X has an exponential distributionwith parameter0 , if its density function is given by可编辑资料 - - - 欢迎下载精品名师归纳总结f xx1 eforx04.5.1可编辑资料 - - - 欢迎下载精品名师归纳总结0forx0Theorem 4.5.1The mean and varia

26、nceof acontinuousrandomvariableX having exponential distribution with parameteris given by可编辑资料 - - - 欢迎下载精品名师归纳总结E X ,D X 2 .可编辑资料 - - - 欢迎下载精品名师归纳总结4.3 Normal Distribution正态分布1. DefinitionThe equationof thenormal probabilitydensity,whose graphisshown in Figure 4.3.1, is可编辑资料 - - - 欢迎下载精品名师归纳总结f x1

27、e x22 / 2 2x可编辑资料 - - - 欢迎下载精品名师归纳总结4.4 Normal Approximation to the Binomial Distribution（二项分布）X Bn, p ,n is large n30, p is close to 0.50,可编辑资料 - - - 欢迎下载精品名师归纳总结可编辑资料 - - - 欢迎下载精品名师归纳总结X Bn, pN np,npq可编辑资料 - - - 欢迎下载精品名师归纳总结4.7 Chebyshev s Theorem（切比雪夫定理）Theorem 4.7.1If a probability distribution

28、has meanand standard deviation, the probability of getting a value which deviates fromby at leastkis at most可编辑资料 - - - 欢迎下载精品名师归纳总结1 . Symbolically ,k2P | X1|k2 .k可编辑资料 - - - 欢迎下载精品名师归纳总结Joint probability distribution（联合分布）Inthestudyofprobability,givenatleasttworandom variables X, Y, ., that are de

29、fined on a probability space,thejointprobabilitydistributionforX,Y,.isaprobabilitydistributionthatgives theprobabilitythat each of X, Y, . falls in any particular range or discrete set ofvalues specified for that variable.5.2 Conditional distribution条件分布Consistent with the definition of conditional

30、probabilityof events when A is the eventX=x and B is the eventY=y, the conditionalprobabilitydistributionofX givenY=yis可编辑资料 - - - 欢迎下载精品名师归纳总结defined aspX x | yp x, yfor allx providedpY y0 .可编辑资料 - - - 欢迎下载精品名师归纳总结pY y5.3 Statistical independent随机变量的独立性Definition5.3.1 Suppose the pair X, Y of real

31、random variables has jointdistributionfunction Fx,y. If the Fx,y obey the product rule可编辑资料 - - - 欢迎下载精品名师归纳总结F x, yFX xFY yfor all x,y.可编辑资料 - - - 欢迎下载精品名师归纳总结the two random variables X and Y are in精d选ep资e料n,d欢en迎t下, o载r the pair X, Y is independent.可编辑资料 - - - 欢迎下载精品名师归纳总结5.4 Covariance and Correl

32、ation协方差和相关系数Definition 5.4.1 Suppose X and Y are random variables. The covariance of the pair X,Y isCov X ,YE XX YY .The correlation coefficient of the pair X, Y is X , YCov X ,Y .XYWhereXE X ,YEY,XD X ,YDY .Definition5.4.2TherandomvariablesXandYare saidtobeuncorrelatediff5.5CoLva wX ,oYfLa0r.ge Nu

33、mbers and Central Limit Theorem中心极限定理We can find the steadily of the frequency of the events inlarge number of random phenomenon. And the average of largenumber of random variables are also steadiness. These resultsare the law of large numbers.Theorem 5.5.1If a sequence Xn : n1 of random variables i

34、sindependent, withE X n , D X n 2,thenWe now define two related quantities whose role in characterizing the interdependence of X and Y we want to examine.可编辑资料 - - - 欢迎下载精品名师归纳总结可编辑资料 - - - 欢迎下载精品名师归纳总结limnP | 1Xnkn k 1|1,for any0 .5.5.1可编辑资料 - - - 欢迎下载精品名师归纳总结可编辑资料 - - - 欢迎下载精品名师归纳总结Theorem 5.5.2Le

35、tnA equals the number of the eventA inn可编辑资料 - - - 欢迎下载精品名师归纳总结Bernoulli trials, andp is the probability of the eventA onany one Bernoulli trial, then可编辑资料 - - - 欢迎下载精品名师归纳总结limnP| nAn|1 for any0 .5.5.2可编辑资料 - - - 欢迎下载精品名师归纳总结可编辑资料 - - - 欢迎下载精品名师归纳总结 频率具有稳固性 Theorem 5.5.3IfX n n1 is independent, wit

36、h可编辑资料 - - - 欢迎下载精品名师归纳总结EX, D X 2, and S*Snn可编辑资料 - - - 欢迎下载精品名师归纳总结nnnn可编辑资料 - - - 欢迎下载精品名师归纳总结thenlimxFn x x for all x.可编辑资料 - - - 欢迎下载精品名师归纳总结population 总体.Definition 6.2.1 A population is the set of data or measurementsconsists of all conceivably possible observations from all objects in a givenApphoepnuolmateionnomn.ay consist of finitely or infinitely manyvarieties.sample （样本、子样）Definition 6.2.2 A sample is a subset of the population from whichsampling 抽样people can draw conclusions about the