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 itself, is certainly an event, which is called a certain event, means that it Salways occurs in the experimen
2、t. 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 trails is denoted by , and if the sequence of relative frequencies nsobtained for larger and large
3、r value of approaches a limit, then this limit is defined as the /snprobability of success in a single trial.“equally likely to occur”-probability（古典概率）If a sample space consists of sample points, each is equally likely to occur. Assume that SNthe event consists of sample points, then the probabilit
4、y that A occurs is Anp()npPAMutually exclusive(互斥事件)Definition 2.4.1 Events are called mutually exclusive, if .12,n , ijijTheorem 2.4.1 If and are mutually exclusive, then B(2.4.1)()()PAPMutually independent 事件的独立性 Two events and are said to be independent if ()()BOr Two events and are independent i
5、f and only ifA.(|)(PBConditional Probability 条件概率The probability of an event is frequently influenced by other events. Definition The conditional probability of , given , denoted by , is defined by BA(|)PBAif . (2.5.1)()(|)PAB()0PThe multiplication theorem 乘法定理If are events, then12k,A121312121()(|)(
6、|)(| )kkPPAAPA If the events are independent, then for any subset k,12,2mii 1212()()()m mPAPAiiii (全概率公式 total probability)Theorem 2.6.1. If the events constitute a partition of the sample space 12,kBS such that for than for any event of ,()0jPBj AS(2.6.2)11()()kkjjjj jAPB（贝叶斯公式 Bayes formula.）Theor
7、em 2.6.2 If the events constitute a partition of the sample space S 12,kBsuch that for than for any event A of S, ,()0jP,j ()0P. for (2.6.2)1()| )iiikjjjPAB1,2ikProof By the definition of conditional probability,()(|)iiPBAUsing the theorem of total probability, we have 1()|(|)iiikjjjPBA1,2ik1. rando
8、m variable definitionDefinition 3.1.1 A random variable is a real valued function defined 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 be a random variable on the sample space . X SThen the function. ()FPxRxis
9、called the distribution function of XNote The distribution function is defined on real numbers, not on ()sample space.3. PropertiesThe distribution function of a random variable has the ()FxXfollowing properties:(1) is non-decreasing.()FxIn fact, if , then the event is a subset of the event 121Xx,th
10、us2Xx1122()()(FxPxxF(2) ,()lim0x.()(3)For any , .This is to say, the R0000li()(xFxFdistribution function of a random variable is right continuous.()FxX3.2 Discrete Random Variables 离散型随机变量Definition 3.2.1 A random variable is called a discrete random variable, if it takes values from a finite set or
11、, a set whose elements can be written as a sequence 12,n geometric distribution （几何分布）X 1 2 3 4 k P p q1p q2p q3pqk1pBinomial distribution（二项分布）Definition 3.4.1 The number of successes in Bernoulli trials is Xncalled a binomial random variable. The probability distribution of this discrete random va
12、riable is called the binomial distribution with parameters and , denoted by .np(,)Bnppoisson distribution（泊松分布）Definition 3.5.1 A discrete random variable is called a Poisson Xrandom variable, if it takes values from the set , and if0,12, ()(;)!kPXkpe 0,12k(3.5.1)Distribution (3.5.1) is called the P
13、oisson distribution with parameter, denoted by .()PExpectation (mean) 数学期望Definition 3.3.1 Let be a discrete random variable. The expectation or Xmean of is defined as(3.3.1)()()xEP2Variance 方差 standard deviation （标准差）Definition 3.3.2 Let be a discrete random variable, having expectation X. Then the
14、 variance of , denote by is defined as the ()EX ()DXexpectation of the random variable 2()X(3.3.6)()DEThe square root of the variance , denote by , is called ()D()DXthe standard deviation of : 12E(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 probab
15、ility density function (,)（概率密度函数）if:(i) ;()0 forany fxR(ii) f(x) is intergrable (可积的) on and .(, )()1fxdDefinition 4.1.2 Let f(x) be a probability density function. If X is a random variable having distribution function , (4.1.1)()()xFxPftdthen X is called a continuous random variable having densit
16、y function f(x). In this case,. (4.1.2)211()(xPxXftd5. Mean（均值）6. variance 方差Similarly, the variance and standard deviation of a continuous random variable X is defined by, (4.1.4)22()()DXEWhere is the mean of X, is referred to as the standard deviation.()EWe easily get. (4.1.5)222()()xfd.4.2 Unifor
17、m Distribution 均匀分布The uniform distribution, with the parameters a and b, has probability density function1for,()0 elswhaxbfxb4.5 Exponential Distribution 指数分布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 defin
18、ed by, (4.1.3)()()Exfdprovided the integral converges absolutely.Definition 4.5.1 A continuous variable X has an exponential distribution with parameter , if its density function is given by(0)(4.5.1)1 for0()0xefTheorem 4.5.1 The mean and variance of a continuous random variable X having exponential
19、 distribution with parameter is given by.2(), ()EXD4.3 Normal Distribution 正态分布1. DefinitionThe equation of the normal probability density, whose graph is shown in Figure 4.3.1, is2()/1()xfxex4.4 Normal Approximation to the Binomial Distribution（二项分布）, n is large (n30), p is close to 0.50,()XBp()(,)
20、Nnq4.7 Chebyshevs Theorem（切比雪夫定理）Theorem 4.7.1 If a probability distribution has mean and standard deviation , the probability of getting a value which deviates from by at least k is at most . Symbolically , 21k.(|)PXJoint probability distribution（联合分布）In the study of probability, given at least two
21、 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 values specified for that variable.5.2Conditional distribu
22、tion 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 asfor all x provided .(,)(|)XYpxy()0Ypy5.3Statistical independent 随机变量的独立性5.4 Covariance and Correlation 协方
23、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.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 average of large nu
24、mber of random variables are also steadiness. These results are the law of large 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 rulefor all x,y.(,)()YFxyythe two random variables X and Y are independent, or the p
25、air X, Y is independent.Definition 5.4.1 Suppose X and Y are random variables. The covariance of the pair X,Y is .Cov(,)()(XYEThe correlation coefficient of the pair X, Y is.Cov(,)()XYWhere (), (), , ().XYXEDDefinition 5.4.2 The random variables X and Y are said to be uncorrelated iff .Cov(,)0number
26、s.Theorem 5.5.1 If a sequence of random variables is :1nXindependent, with 2(), (), nnEDthen. (5.5.1)1lim(|)1, forany 0nknPXTheorem 5.5.2 Let equals the number of the event A in n Bernoulli Atrials, and p is the probability of the event A on any one Bernoulli trial, then. (5.5.2)lim(|)1 forany 0AnP(
27、频率具有稳定性)Theorem 5.5.3 If is independent, with()nX2*, (), andnnnSEDthen .lim() foral nxFxpopulation (总体 ).A population may consist of finitely or infinitely many varieties. sample （样本、子样）Definition 6.2.1 A population is the set of data or measurements consists of all conceivably possible observations
28、 from all objects in a given phenomenon. sampling(抽样)taking a sample: The process of performing an experiment to obtain a sample from the population is called sampling. 中位数Sample Distributions 抽样分布1sampling distribution of the mean 均值的抽样分布Theorem 6.3.1 If is mean of the random sample of size X nX,21
29、from a random variable which has mean and the variance , then2and .)(EnXD2)(It is customary to write as and as . X2XHere, is called the expectation of the mean.均值的期望()EXis called the standard error of the mean. 均值的标准差nDefinition 6.2.2 A sample is a subset of the population from which people can draw
30、 conclusions about the whole.Definition 6.2.4 If a random sample has the order statistics , then)()2()1,nX(i) The Sample Median is evnisf21od)12()()1(0XMn(ii) The Sample Range is .)1()(Rn7.1 Point Estimate 点估计Definition 7.1.1 Suppose is a parameter of a population, is a nX,1random sample from this p
31、opulation, and is a statistic that is ),(1nXTa function of . Now, to the observed value , if we use nX,1 nx,1as an estimated value of , then is called a point ),(1nxT ),(1estimator of and is referred as a point estimate of . The ),(1nxT point estimator is also often written as .Unbiased estimator(无偏
32、估计量 )Definition 7.1.2. Suppose is an estimator of a parameter . Then is unbiased if and only if .)(Eminimum variance unbiased estimator（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 have,)(Dthen is called the minimum variance unbi
33、ased estimator of . Sometimes it is also called best unbiased estimator.3. Method of Moments 矩估计的方法Definition 7.1.4 Suppose constitute a random sample from the nX,21population X that has k unknown parameters . Also, the k,21population has firs k finite moments that depends )(,)(,XEXEon the unknown p
34、arameters. Solve the system of equations, (7.1.4)nikikniiiiXEXE1122)()(to get unknown parameters expressed by the observations values, i.e.for . Then is an estimator of by ),(21kjj Xkj,2 j jmethod of moments. Definition7.2.1 Suppose that is a parameter of a population, nX,1is a random sample of from
35、 this population, and and ),(1Tare two statistics such that . If for a given with ),(12nXT 21, we have0.)(21PThen we refer to as a confidence interval for . ,21%0Moreover, is called the degree of confidence. and are called 12lower and upper confidence limits. The estimation using confidence interval
36、 is called interval estimation. confidence interval- 置信区间 lower confidence limits- 置信下限 upper confidence limits- 置信上限degree of confidence-置信度2极大似然函数 likelihood functionDefinition 7.5.1 A random sample has the observed values nx,21from a population with an unknown parameter . Then the likelihood func
37、tion for this sample is );,()21nxfLin which is defined in (7.5.1).;,(21nxfmaximum likelihood estimate （最大似然估计）Definition 7.5.2 If there is a value such that)(Lfor all , then is called a maximum likelihood estimate of . 8.1 Statistical Hypotheses（统计假设） Definition 8.1.1 A statistical hypothesis is an
38、assertion or conjecture concerning one or more population.Definition 8.1.2 A statistical procedure or decision rule that leads to establishing the truth or falsity of a hypothesis is called a statistical test.显著性水平Definition 8.2.1 Which is called significant level, describes how far the sample mean
39、is far from the population mean.Two Types of ErrorsDefinition 8.2.3 If it happens that the hypothesis being tested is actually true, and if from the sample we reach the conclusion that it is false, we say that a type I error has been committed.Definition 8.2.4 If it happens that hypothesis being tested is actually false, and if from the sample we reach the conclusion that it is true, we say that a type II error has been committed.
