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 certainly an event, which is called a certain event, means that it always occurs in the experiment

2、.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 experiment.Probability of events (概率)If the number of successes in n trails is denoted by s, and if the sequence of relative frequencies s/ n obtained for larger and

3、larger value of n approaches a limit, then this limit is defined as the probability of success in a single trial.a 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 that the event A consists of n sample points, then the

4、 probability p that A occurs isP(A)Mutually exclusive(互斥事件)Definition 2.4 .1 Events A,A2,L , An are called mutually exclusive , if Ai I Aj, i jTheorem 2.4.1 If A and B are mutually exclusive, thenP(AU B) P(A) P(B)(2.4.1)Mutually independent事件的独立性Two events A and B are said to be independent ifP(AI B

5、) 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 event is frequently influenced by other events.Definition The conditional probability of B , given A, denoted by P( B | A), is defined byP(AI B)P(B| A)P(AJif P(A) 0.(2.5.1)Th

6、e multiplication theorem乘法定理If A,A2,L ,Ak are events, thenP(AiIA2ILAk)P(A) P(A2|A) P(A3|A IA2)L P(A|AiIA2IL IAk1)If the events A1,A2,L ,Akare independent, then for any subseti1,i2,L ,im1,2,L ,k,P(A I A I L A ) P(A )P(A )L P(A )i 1 i 2imi 1i 2i m(全概率公式total probability)Theorem 2.6.1. If the events B1

7、,B2,L , Bk constitute a partition of the samplespace S such that P(Bj) 0 for j 1,2,L ,k, than for any event A of S,P(A) P(AI Bj)P(Bj )P(AI Bj)(2.6.2)(贝叶斯公式 Bayes; formula.)Theorem 2.6.2 If the events B1, B2,L , Bk constitute a partition of the sample spaceS such that P(Bj) 0 for j 1,2,L ,k, than for

8、 any event A of S,P(A) 0,P(Bi|A)旧尸网P(Bj)P(A|Bj) j 1fori 1,2,L ,k(2.6.2)Proof By the definition of conditional probability,Using the theorem of total probability, we havei 1,2,L ,kP(BJA)J旧)P(A|Bi)P(Bj)P(A|Bj) j 1random variable definitionDefinition 3.1.1 A random variable is a real valued function de

9、fined on a sample space; i.e. it assigns a real number to each sample point in the sample space.Distribution functionDefinition 3.1.2 Let X be a random variable on the sample space S. Then the functionF(X) P(X x) .x Ris called the distribution function of XNote The distribution function F(X) is defi

10、ned on real numbers, not on sample space.PropertiesThe distribution function F(x) of a random variable X has the following properties:F(x) is non-decreasing.In fact, if x1 x2, then the event X x1 is a subset of the event X x2 ,thusF(Xi) P(X Xi) P(X x2) F(x2)(2)F( ) Xim F(x) 0,F( ) Hm F(x) 1.(3)For a

11、ny x0 R, lim F(x) F(% 0) F(x0) .This is to say, the x M 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 random variable, if it takes values from a finite set or, a set whose element

12、s can be written as a sequence a1,a2,L an,L geometric distribution (几何分布)X1234kPpq f (x) 0 for any x R;(ii) f(x) is intergrable (可积的)on (,) and f (x)dx 1. pq2pq3pqk 一1pBinomial distribution(二项分布)Definition 3.4.1 The number X of successesin n Bernoulli trials is called a binomial random variable. The

13、 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 Poisson random variable, if it takes values from the set 0,1, 2,l , and if k

14、 P(X k) p(k; ) e ,0k 0,1,2,Lk!.5.1)Distribution (3.5.1) is called the Poisson distribution with parameter , 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 asE(X) xP(X x)(3.3.1)xVariance 方差 standard deviation (标准差)Def

15、inition 3.3.2 Let X be a discrete random variable, having expectation E(X) . Then the variance of X , denote by d(x) is 22tl、2defined as the expectation of the random variable (X )D(X) E (X)2(3.3.6)The square root of the variance D(X), denote by , D(X) , is1 called the standard deviation of X : d(x)

16、 e x 2 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)Definition 4.1.2Let f(x) be a probability density function. If X is a random variable having distribution functionxF(x) P(X x) f dt,(4.1.1)then X

17、is called a continuous random variable having density function f(x). In this case,x2P(Xi Xx2)f (t)dt.(4.1.2)x1Mean (均值)Definition 4.1.2 Let X be a continuous random variable having probability density function f(x). Then the mean (or expectation) of X is defined byE(X) xf(x)dx,(4.1.3)variance 方差Simi

18、larly, the variance and standard deviation of a continuous random variable X is defined by2_2D(X) E(X ) ),(4.1.4)Where E(X) is the mean of X, is referred to as the standard deviation .We easily get2 D(X)x2f(x)dx2. (4.1.5)4.2 Uniform Distribution均匀分布The uniform distribution, with the parameters a and

19、 b, has probability density function1 f. for a x b, f (x) b a0 elsewhere,4.5 Exponential Distribution指数分布Definition 4.5.1 A continuous variable X has an exponential distribution with parameter(0), if its density function is given byf(x)for(4.5.1)0 for x 0Theorem 4.5.1 The mean and variance of a cont

20、inuous random variable X having exponential distribution with parameter is given byE(X) , D(X) 2.Normal Distribution正态分布1. DefinitionThe equation of the normal probability density, whose graph is shown in Figure 4.3.1, isf(x)Normal Approximation to the Binomial Distribution (二项分布)X B(n, p), n is lar

21、ge (n30), p is close to 0.50,X B(n, p) N(np, npq)Chebyshev s Theo rem 匕雪夫定理)Theorem 4.7.1 If a probability distribution has mean 科 and standard deviation a the probability1 of getting a value which deviates from (iby at least k (ris at most - . Symbolically , k1P(|X | k )2.kJoint probability distrib

22、ution(联合分布)In the study of probability, given at least two random variables X, Y,.,that are defined on a probability space, the joint probability distribution 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 va

23、lues specified for that variable.5.2Conditional distribution条件分布Consistent with the definition of conditional probability of events when A is the event X=x and B is the event Y=y, the conditional probability distribution of X given Y=y is defined as ,、p(x, y) pX(x|y)for all x provided pY(y) 0.Py(y)5

24、.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 distribution function F(x,y). If the F(x,y) obey the product ruleF(x, y) Fx(x)FyH) for all x,y.the two random variables X and Y are independent, or the pair X, Y is independent.5.4 Covariance

25、and Correlation 协方差和相关系数We now define two related quantities whose role in characterizing the interdependence of X and Y we want to examine.Definition 5.4.1 Suppose X and Y are random variables. The covariance of the pair X,Y isCov(X,Y) E(X x)(Y y).The correlation coefficient of the pair X, Y is(X,

26、Y)Cov(X,Y)X YWhere x E(X), 丫 E(Y), x ,D(X), y , D(Y).Definition 5.4.2 The random variables X and Y are said to be uncorrelated iffCov(X,Y) 0.5.5 Law of Large Numbers and Central Limit Theorem 中心极限定理We can find the steadily of the frequency of the events in large number of random phenomenon. And the

27、average of large number of random variables are also steadiness. These results are the law of large numbers.Theorem 5.5.1 If a sequence Xn: n 1 of random variables is independent, withE(Xn), D(Xn)2,then_ 1 nlim P(| Xk |)1, for any 0.(5.5.1)nn k 1Let nA equals the number of the event A in n Bernoulli

28、trials, and p is the probability of the event A on any one Bernoulli trial, thenlim P(| A| ) 1 for any 0.(5.5.2)n nis independent, withXn(n 1)(频率具有稳定性IfE(Xn),D(Xn), and Sn -“-：=then Xim Fn(x)(x) for allx.population (总体)Definition 6.2.1 A population is the set of data or measurements consists of all

29、conceivably possible observations from all objects in a given phenomenon.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 whichsampinaKdfc Ipeoonclusions about the whole.taking a sample: The process of perfor

30、ming an experiment to obtain a sample from the population is called sampling.中位数Definition 6.2.4 If a random sample has the order statisticsx(i),x(2), ,X(n), thenThe Sample Median isX(n 1) 2if n is oddMo 1 2X(n) 2(21)if n is evenThe Sample RangeisX (n) X (1).Sample Distributions 抽样分布1. sampling dist

31、ribution of the mean 均值的抽样分布Theorem 6.3.1 If X is mean of the random sampleX1,X2, ,Xn of sizen from a random variable X which has mean and the variance2, then2 e（x） and d（x）. nIt is customary to write e（x） as X and d（x） as ：.Here, e（x） is called theexpectation of the mean .均值的期望X -= is called the st

32、andard error of the mean. 均值的标准差 , n7.1 Point Estimate 点估计Definition 7.1.1 Suppose | is a parameter of a population, x， ,Xn is a random sample from this population, and t（x1， ,Xn） is a statistic that is a function of 卜h Xn. Now, to the observed value x1， , xn, if we use T（x ，xn） as an estimated valu

33、e of |, then T（X1， ，Xn） is called a point estimator of and T（x1，is referred as apoint estimate of.The point estimator is also often written as ?.Unbiased estimator优偏估计量）Definition 7.1.2. Suppose ? is an estimator of a parameter . Then ? is unbiased if and only ifE（?）.minimum variance unbiased estima

34、tor （最小方差无偏估计量）Definition 7.1.3 Let ? be an unbiased estimator of . If for any ? which is also an unbiased estimator of , we haveD（?） D（7）,then ? is called the minimum variance unbiased estimator of .Sometimes it is also calledbest unbiased estimator.3. Method of Moments 矩估计的方法Definition 7.1.4Suppos

35、e X1,X2, ,xn constitute a random sample from the population X that has k unknown parameters 1, 2, , k. Also, the population has firs k finite moments e(x), E(X2), ,E(Xk) that depends on the unknown parameters. Solve the system of equationsn E(X) Xi n i 1nE(X2)1 Xi(7.1.4)n i 1 nE(Xk)1 Xin i 1to get u

36、nknown parameters expressed by the observations values, i.e.j ?j(X1,X2, ,XJ for j 1,2, ,k. Then ? is an estimator of j by method of moments.Definition7.2.1 Suppose that is a parameter of a population, X1, ,Xnis a random sample of from this population, and? T1(X1, ,Xn) and ?2 T2(X1, ,Xn) are two stat

37、istics such that ?2. If for a given with 01, we havep(?2) 1.Then we refer to Z, ?2 as a (1)100% confidence interval for .Moreover, 1 is called the degree of confidence. ? and ?2 are called lower and upper confidence limits. The estimation using confidence interval is calledinterval estimation .confidence interval置信区间lower confidence limits信下限upper confidence limits 置信上限degre