概率论与数理统计英文版总结

1、Sample Space样本空间The set of all possible outcomes of a statistical experiment is called the sample space.Event事件An event is a subset of a sample space.certain event (必然事件)：The sample space S itself, is certa inly an eve nt, which is called a certa in eve nt, means that it always occurs in the experim

2、e nt.impossible event(不可能事件)：The empty set, denoted by ；” , is also an event, called an impossible event, means that it never occurs in the experime nt.Probability of events (概率)If the number of successes in n trails is denoted by s , and if the sequenee of relative frequencies s/ n obtained for lar

3、ger and larger value of n approaches a limit, then this limit is defined as the probability of success in a single trial.“equally likely to occur-probability (古典概率)If a sample space S consists of N sample points, each is equally likely to occur. Assume thatthe event A consists of n sample points, th

4、en the probability p that A occurs isn,P(AJMutually exclusive(互斥事件)Definition 2.4 . 1 Events Ai, A2)1, An are called mutually exclusive, if All Aj =-_ jjTheorem 2.4.1 If A and B are mutually exclusive, thenP(aUb)二 P(A) P(B)(2.4.1)Mutually independent 事件的独立性Two events A and B are said to be independe

5、nt ifP(A“B)二 P(A) P(B)Or Two events A and B are independent if and only ifP(B| A) =P(B).Conditional Probability 条件概率The probability of an eve nt is freque ntly in flue need by other eve nts.Definition The conditional probability of B , give n A, de no ted by P(B|A), is defined byp(aDb)P(B| A)if P(A)

6、 0.(2.5.1)P(A)The multiplication theorem 乘法定理If A,A2,|(,A are events, thenP(A1 C1A2 仃Ak) =P(A) P(A2|A1)P(A3|AriA2)P(Ak|A1plA2nn Ak)If theeventsA1,A2( ,Akare independent, then for any subseti1.i2JH.im1,2l|,k,p(A na na)=pa pa )p)i 1i 2i mi 1i 2i m(全概率公式total probability)Theorem 2.6.1. If the events B1

7、,B2|, Bk constitute a partition of the sample space S such that P(Bj)=O for j -1,2l,k, than for any event A of S,kkP(A)八 P(AClBj)八 P(Bj)P(AClBj)(2.6.2)jj#(贝叶斯公式Bayes formula.)Theorem 2.6.2 If the events B1, B2|, Bk constitute a partition of the sample space S such that P(Bj)0 for j =1,2l,k, tha n fo

8、r any eve nt A of S, P(A) =0,P(Bi |A 卜厂 FAB ). for i =12 川,k(2.6.2)瓦 P(Bj 尸 A Bj )j=1Proof By the definition of conditional probability,P(Bi |A)二P(A)Using the theorem of total probability, we haveP |A 上 kP(Bi 尸 A Bi ) i , 21k,Z P )P A Bj )j 41. random variable definitionDefinition 3.1.1 A random var

9、iable is a real valued fun cti on defi ned on a sample space; i.e. it assigns a real number to each sample point in the sample space.2. Distribution functionDefinition 3.1.2 Let X be a random variable on the sample space S.Then the fun cti onF(X) =P(X 乞 x).x Ris called the distribution function of X

10、Note The distribution function F(X) is defined on real numbers, not on sample space.3. PropertiesThe distribution function F(x) of a random variable X has the followi ng properties:(1) F(x) is non-decreasing.In fact, if x X2, the n the eve nt X x is a subset of the eve ntX X2,thusF(xJ = P(X : xj - P

11、(X : x2) = F(x2)(2) F(：)二 lim F(x) =0,F( ：) = l i rfi x()J +oc(3) For any x R, lim F(x)二 F(x 0) = F(x) .This is to say, theX /o 0distribution function F (x) of a random variable X is right continuous.3.2 Discrete Random Variables 离散型随机变量Definition 3.2.1 A random variable X is called a discrete rando

12、m variable, if it takes values from a finite set or, a set whose elements can be writte n as a seque nce a1,a2,ll(anllgeometric distribution (几何分布)X1234kPp1q p2q p3q pqk1pBinomial distribution (二项分布)Definition 3.4.1 The number X of successesin n Bernoulli trials is called a binomial random variable.

13、 The probability distribution of this discrete random variable is called the binomial distribution with parameters n and p, denoted by B(n, p).poisson distribution(泊松分布)Definition 3.5.1 A discrete random variable X is called a Poissor random variable, if it takes values from the set 0,1, 2,111, and

14、if? k _P(X =k) = p(k;町=, 人0k = 0,1,2,lllk!(3.5.1)Distributen(3.5.1) is called the Poisson distribution withparameter扎，denoted by P.Expectation (mean)数学期望Definition 3.3.1 Let X be a discrete random variable. The expectation or mean of X is defined asA = E(X)=I： xP(X =x)(3.3.1)x2. Variance 方差 standard

15、 deviation (标准差)Definition 3.3.2 Let X be a discrete random variable, having expectation e(x)=4. Then the variance of X , denote by d(x)is defined as the expectation of the random variablex -卩)D(X) =E(X -叮)(3.3.6)The square root of the varianee D(X), denote by Jd(x), is1called the standard deviation

16、 of X : Jd(x)=(E(x 卩)2(3.3.7)probability density function 概率密度函数Definition 4.1.1 A function f(x) defined on (-：) is called a probability density function (概率密度函数)if:(i) f (x) 一0 for any x R;C3O(ii) f(x) is in tergrable (可积的)on (- 二)a nd f (x)dx = 1.Definition 4.1.2Let f(x) be a probability density f

17、unction. If X is a random variable having distribution functionF(x)二 P(X ex)二 f(t)dt,(4.1.1)-jodthen X is called a continuous random variable having density function f(x). In this case,X2P(m : X ：X2)= f (t)dt.(4.1.2)X15. Mean (均值)Definition 4.1.2 Let X be a continuous random variable having probabil

18、ity density function f(x). Then the mean (or expectation) of X is defined byE(X)二.xf (x)dx ,(4.1.3)-oO6. variance 方差Similarly, the variance and standard deviation of a continuous random variable X is defined by22二 二 D(X)二 E(X - J ),(4.1.4)Where J = E(X) is the mean of X, - is referred to as the stan

19、dard deviation.We easily getoO；2=D(X)=X2f(x)dx-S2. (4.1.5)-oO4.2 Uniform Distribution 均匀分布The uniform distributi on, with the parameters a and b, has probability den sity fun cti on工1for a x : b,f(x)二 b-a0elsewhere,4.5 Exponential Distribution 指数分布Definition 4.5.1 A continuous variable X has an expo

20、nential distribution with parameter(/ 、0), if its density function is given byf (x)=1-xe for0 forx 0x _0(4.5.1)Theorem 4.5.1 The mean and variance of a continuous random variable X having exponential distributi on with parameter is give n by2E(X)二,D(X)二4.3 Normal Distribution正态分布1. DefinitionThe equ

21、ati on of the no rmal probability den sity, whose graph is show n in Figure 4.3.1, isf(x)4X-J2/2；：2-:x :4.4 Normal Approximation to the Binomial Distribution (二项分布)X B(n, p), n is large (n30), p is close to 0.50,X B(n, p) : N (np, npq)4.7 Chebyshev s Theory切比雪夫定理)Theorem 4.7.1 If a probability distr

22、ibution has meanand standard deviation qthe probability of1 getting a value which deviates from (iby at least k Hs at most _2 . Symbolically , k1P(|X|一2)乞kJoint probability distribution (联合分布)In the study of probability, given at least two random variables X,Y, ., that are defined on a probability s

23、pace, the joint probability distributen for X, Y, . is a probability distribution that gives the probability that each of X, Y, . falls in any particular range or discrete set of values specified for that variable.5.2Conditional distribution 条件分布Consistent with the definition of conditional probabil

24、ity of events whe n A is the eve nt X=x and B is the eve nt Y=y, the con diti onal probability distributi on of X give n Y=y is defi ned aspX(x| y) = p(x, y) for all x provided pY(y) = o.py(y)5.3Statistical independent 随机变量的独立性Definition 5.3.1 Suppose the pair X, Y of real random variables has joint

25、 distribution functionF(x,y). If the F (x,y) obey the product ruleF (x, y) =Fx(x)FY(y) for all x,y.the two random variables X and Y are independent, or the pair X, Y is independent.5.4 Covariance and Correlation 协方差和相关系数We now defi ne two related qua ntities whose role in characteriz ing the in terd

26、epe ndence ofX and Y we want to exam ine.Definition 5.4.1 Suppose X and Y are random variables. The covariance of the pair X,Y isCov(X,Y) =E(Xx)(Yy)The correlation coefficient of the pair X, Y is(X,Y“沁卫Where 呎二 E(X),亠二 E(Y), JD(X)，二丫 - D(Y).Definition 5.4.2 The random variables X and Y are said to b

27、e uncorrelated 肝5.&僭如%f Lage Numbers and Central Limit Theorem_中心极限定We can find the steadily of the frequency of the events in large number of random phenomenon. And the average of large number of ran dom variables are also stead in ess.These results are the law of largenu mbers.Theorem 5.5.1If a se

28、quenee xn: n_i of random variables isin depe ndent, withE(Xn)D(Xn) L,the nlim P(| Z Xk#|E)=1, for any 0 . nnz(5.5.1)Theorem 5.5.2 Let nA equals the number of the eventA in n Bernoullitrials, and p is the probability of the eve nt A on any one Berno ulli trial,the nlim P(| 耳一卩 名)=1 for any s 0. Y n(频

29、率具有稳定性)Theorem 5.5.3 If Xn(n 1) is independent, with112* Sn n,LJ_E(Xn) = P, D(Xn) , and Sn =rn(5.5.2)then lim_Fn(x)=：(x) for all x.population (总体)Definition 6.2.1 A population is the set of data or measurements.eonsists of all eoneeivably possible observations from all objects in a give n phe nomeno

30、n.A population may consist of finitely or infinitely many varieties.sample （样本、子样）Definition 6.2.2 A sample is a subset of the population from whichn clusi ons about the whole.sampling抽样)people caff draw co taking a sample: The process of performing an experiment to obtain aSample from the populatio

31、n is calledsampling中位数Definition 6.2.4 If a random sample has the order statisticsx（i）,x（2）, ,x（n）, then(i) The Sample Median is1XnXn2 IL(2)(2 1)if n is oddif n is even(ii) The Sample RangeisR = X (n)- X (I).Sample Distributions 抽样分布1. sampling distribution of the mean均值的抽样分布Theorem 6.3.1 If x is me

32、an of the random sampleX1,X2 - ,Xn of size n from a random variable X which has mean J and the varianee 二,the n2 Qe（x）- and d（x）=nIt is customary to write e（x） as 咲 and d（x）as ；r.Here, E（X）= i is called the expectation of the mean匀值的期望is called the standard error of the mean 均值的标准差x .n7.1 Point Esti

33、mate 点估计Definition 7.1.1 Suppose is a parameter of a population, ，xn is a random sample from this population, and t（x“ ,xn）is a statistic that is a function of x n. Now, to the observed value , xn, if we use T（X1, ,xn）as an estimated value of then t（x ，Xn） is called a point estimator of 0 and t（x ，x

34、n）is referred as a point estimateof 目 The point estimator is also ofte n writte n as ?Un biased estimator无 偏估计量）Definition 7.1.2. Suppose is an estimator of a parameter . Then2 is unbiasedif and only ifE（国- Aminimum variance unbiased estimator 最小方差无偏估计量）Definition 7.1.3 Let 2 be an unbiased estimato

35、r of . If for any ? which is also an unbiased estimator of , we haveD（闵兰 D（&）,then 彳 is called the minimum variance unbiased estimator of -.Sometimes it is also calledbest unbiased estimator3. Method of Moments矩估计的方法Definition 7.1.4Suppose Xi,X2, ,Xn constitute a random sample from thepopulation X t

36、hat has k unknown parameters “，七,入.Also, the population has firs k finite moments e(x), e(x2), ,E(xk)that depends on the unknown parameters. Solve the system of equati ons1 nE(X)Xin z1 n(7.1.4)E(X2) = Xi2n znE(Xk)二、Xikn yto get unknown parameters expressed by the observati ons values, i.e.9?(X1, X2-

37、 ,xk) for j=1,2,，k. The n is an estimator of 日 j by method of moments.Definition7.2.1 Suppose that is a parameter of a population,X1, ,Xn is a random sample of from this population, and冈=(X1，Xn) and 超=T2(Xj，Xn) are two statistics such that闵 eg. If for a given a with d, we haveThen we refer to &, as a (1)100% confidence interval for 日.Moreover, 1 ： is called the degree of confidence 气 and 兔 are called lower and upper confidence limits. The estimation using con fide nee in t