概率论与数理统计英文版总结电子教案

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概率论与数理统计(英文) 第九章

9. Nonparametric Statistics9.1 Sign Test 符号检验1The simplest of all nonparametric methods is the sign test, which is usually used to test the significance of the difference between two means in a paired experiment.最简单的非参数检验是符号检验检验两个总体均值差的显著程度It is particularly suitable when the various pairs are observed under different conditions, a case in which the assumption of normality may not hold. However, because of its simplicity, the sign test is often used even though the populations are normally distributed. As is implied by its name in this test only the sign of the differencebetween the paired variates is used.若两个总体的均值相等，那么符号‘＋’、‘－’的概率一样。

D = sign of (X 1－X 2 )If p denotes the probability of a difference D being positive andq the probability of its being negative, we have as hypothesis p=1/2. appropriate test statistic is X , X~B (n, p)， X --- N(‘+”)we will reject 0Hin favor of1Honly if the proportion of plussigns is sufficiently less than 1/2, that is , when the value x of our random variable is small. Hence, if the computed P -value12()P P X x when p =≤=is less than or equal to the significance level α, we reject 0Hinfavor of1H .we reject0Hin favor1Hwhen the proportion of plus signs issignificantly less than or significantly greater than 1/2. This, of course, is equivalent to x being sufficiently small or sufficiently large, respectively. Therefore, if /2x n < and the computed P-value 122()P P X x when p =≤=is less than or equal to α, or if /2x n > and the computed P-value 122()P P X x when p =≥= is less than or equal to α, we reject 0Hin favor1H .Car Radial tires Belted tires D1 4.2 4.1 ＋ 2 4.7 4.9 －3 6.6 6.2 ＋4 7.0 6.9 ＋5 6.7 6.8 －6 4.5 4.4 ＋7 5.7 5.78 6.0 5.8 ＋9 7.4 6.9 ＋10 4.9 4.911 6.1 6.0 ＋12 5.2 4.9 ＋13 5.7 5.3 ＋14 6.9 6.5 ＋15 6.8 7.1 －16 4.9 4.8 ＋符号检验的利弊n 必须比较大因为对于n =5的样本，会出现永远不拒绝“总体均值相等“的假设。

概率论与数理统计英文版总结

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、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 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、Assumethat the event A consists of n sample points, then the probability p that A occurs is()np P AN==Mutually exclusive(互斥事件)Mutually independent 事件的独立性Two events A and B are said to be independent if()()()P A B P A P B=⋅IOr Two events A and B are independent if and only if(|)()P B A P B=、Conditional Probability 条件概率The probability of an event is frequently influenced by other events、If 12k ,,,A A A L are events, then12k 121312121()()(|)(|)(|)k k P A A A P A P A A P A A A P A A A A -=⋅⋅I I L I L I I L I If the events 12k ,,,A A A L are independent, then for any subset 12{,,,}{1,2,,}m i i i k ⊂L L ,1212()()()()m mP A A A P A P A P A i i i i i i =I I L L (全概率公式 total probability)()(|)()i i P B A P B A P A =IUsing the theorem of total probability, we have1()(|)(|)()(|)i i i k j jj P B P A B P B A P B P A B ==∑ 1,2,,i k =L1、 random variable definition2、 Distribution functionNote 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 following properties:3、2 Discrete Random Variables 离散型随机变量geometric distribution (几何分布)Binomial distribution(二项分布)poisson distribution(泊松分布)Expectation (mean) 数学期望2.Variance 方差standard deviation (标准差)probability density function概率密度函数5、 Mean(均值)6、 variance 方差、4、2 Uniform Distribution 均匀分布The uniform distribution, with the parameters a a nd b , has probability density function1 for ,()0 elsewhere,a xb f x b a ⎧<<⎪=-⎨⎪⎩4、5 Exponential Distribution 指数分布4、3 Normal Distribution 正态分布1、Definition4、4 Normal Approximation to the Binomial Distribution(二项分布)4、7 Chebyshev’s Theorem(切比雪夫定理)Joint 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 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、5.2C onditional 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(,)(|)()X Y p x y p x y p y = for all x provided ()0Y p y ≠、 5.3 S tatistical independent 随机变量的独立性interdependence of X and Y we want to examine 、number of random phenomenon 、 And the average of large number of random variables are also steadiness 、 These results are the law of large numbers 、population (总体)sample (样本、子样)中位数It is customary to write )(X E as X μ and )(X D as 2X σ、Here, ()E X μ= is called the expectation of the mean 、均值的期望 n X σσ= is called the standard error of the mean 、均值的标准差 7、1 Point Estimate 点估计Unbiased estimator(无偏估计量)minimum variance unbiased estimator(最小方差无偏估计量)3、Method of Moments 矩估计的方法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 function显著性水平Two Types of Errors。

概率论与数理统计英文版总结

概率论与数理统计英文版总结Probability theory and mathematical statistics are essential branches of mathematics that deal with the study and analysis of uncertain events and data. These two fields are closely related and provide the foundation for making informed decisions and drawing conclusions based on probability and statistical analysis. In this summary, we will explain the key concepts and principles of probability theory and mathematical statistics.Probability theory is concerned with the study of random events and their likelihood of occurrence. It is used toquantify uncertainty and provide a framework for making predictions and decisions in various disciplines, including natural sciences, social sciences, finance, and engineering. The fundamental concept in probability theory is the probability of an event, which is a value between 0 and 1 that represents the likelihood of the event occurring.Probability theory is built upon three main axioms:2. The probability of the entire sample space is always 1.。

概率论与数理统计英文文献

Introduction to probability theory andmathematical statisticsThe theory of probability and the mathematical statistic are carries on deductive and the induction science to the stochastic phenomenon statistical rule, from the quantity side research stochastic phenomenon statistical regular foundation mathematics discipline, the theory of probability and the mathematical statistic may divide into the theory of probability and the mathematical statistic two branches. The probability uses for the possible size quantity which portrays the random event to occur. Theory of probability main content including classical generally computation, random variable distribution and characteristic numeral and limit theorem and so on. The mathematical statistic is one of mathematics Zhonglian department actually most directly most widespread branches, it introduced an estimate (rectangular method estimate, enormousestimate), the parameter supposition examination, the non-parameter supposition examination, the variance analysis and the multiple regression analysis, the fail-safe analysis and so on the elementary knowledge and the principle, enable the student to have a profound understanding tostatistics principle function. Through this curriculum study, enables the student comprehensively to understand, to grasp the theory of probability and the mathematical statistic thought and the method, grasps basic and the commonly used analysis and the computational method, and can studies in the solution economy and the management practice question using the theory of probability and the mathematical statistic viewpoint and the method.Random phenomenonFrom random phenomenon, in the nature and real life, some things are interrelated and continuous development. In the relationship between each other and developing, according to whether there is a causal relationship, very different can be divided into two categories: one is deterministic phenomenon. This kind of phenomenon is under certain conditions, will lead to certain results. For example, under normal atmospheric pressure, water heated to 100 degrees Celsius, is bound to a boil. This link is belong to the inevitability between things. Usually in natural science is interdisciplinary studies and know the inevitability, seeking this kind of inevitable phenomenon.Another kind is the phenomenon of uncertainty. This kind of phenomenon is under certain conditions, the resultis uncertain. The same workers on the same machine tools, for example, processing a number of the same kind of parts, they are the size of the there will always be a little difference. As another example, under the same conditions, artificial accelerating germination test of wheat varieties, each tree seed germination is also different, there is strength and sooner or later, respectively, and so on. Why in the same situation, will appear this kind of uncertain results? This is because, we say "same conditions" refers to some of the main conditions, in addition to these main conditions, there are many minor conditions and the accidental factor is people can't in advance one by one to grasp. Because of this, in this kind of phenomenon, we can't use the inevitability of cause and effect, the results of individual phenomenon in advance to make sure of the answer. The relationship between things is belong to accidental, this phenomenon is called accidental phenomenon, or a random phenomenon.In nature, in the production, life, random phenomenon is very common, that is to say, there is a lot of random phenomenon. Issue such as: sports lottery of the winning Numbers, the same production line production, the life of the bulb, etc., is a random phenomenon. So we say: randomphenomenon is: under the same conditions, many times the same test or survey the same phenomenon, the results are not identical, and unable to accurately predict the results of the next. Random phenomena in the uncertainties of the results, it is because of some minor, caused by the accidental factors.Random phenomenon on the surface, seems to be messy, there is no regular phenomenon. But practice has proved that if the same kind of a large number of repeated random phenomenon, its overall present certain regularity. A large number of similar random phenomena of this kind of regularity, as we observed increase in the number of the number of times and more obvious. Flip a coin, for example, each throw is difficult to judge on that side, but if repeated many times of toss the coin, it will be more and more clearly find them up is approximately the same number.We call this presented by a large number of similar random phenomena of collective regularity, is called the statistical regularity. Probability theory and mathematical statistics is the study of a large number of similar random phenomena statistical regularity of the mathematical disciplines.The emergence and development of probability theoryProbability theory was created in the 17th century, it is by the development of insurance business, but from the gambler's request, is that mathematicians thought the source of problem in probability theory.As early as in 1654, there was a gambler may tired to the mathematician PASCAL proposes a question troubling him for a long time: "meet two gamblers betting on a number of bureau, who will win the first m innings wins, all bets will be who. But when one of them wins a (a < m), the other won b (b < m) bureau, gambling aborted. Q: how should bets points method is only reasonable?" Who in 1642 invented the world's first mechanical addition of computer.Three years later, in 1657, the Dutch famous astronomy, physics, and a mathematician huygens is trying to solve this problem, the results into a book concerning the calculation of a game of chance, this is the earliest probability theory works.In recent decades, with the vigorous development of science and technology, the application of probability theory to the national economy, industrial and agricultural production and interdisciplinary field. Many of applied mathematics, such as information theory, game theory, queuing theory, cybernetics, etc., are based on the theory of probability.Probability theory and mathematical statistics is a branch of mathematics, random they similar disciplines are closely linked. But should point out that the theory of probability and mathematical statistics, statistical methods are each have their own contain different content.Probability theory, is based on a large number of similar random phenomena statistical regularity, the possibility that a result of random phenomenon to make an objective and scientific judgment, the possibility of its occurrence for this size to make quantitative description; Compare the size of these possibilities, study the contact between them, thus forming a set of mathematical theories and methods.Mathematical statistics - is the application of probability theory to study the phenomenon of large number of random regularity; To through the scientific arrangement of a number of experiments, the statistical method given strict theoretical proof; And determining various methods applied conditions and reliability of the method, the formula, the conclusion and limitations. We can from a set of samples to decide whether can with quite large probability to ensure that a judgment is correct, and can control the probability of error.- is a statistical method provides methods are used in avariety of specific issues, it does not pay attention to the method according to the theory, mathematical reasoning.Should point out that the probability and statistics on the research method has its particularity, and other mathematical subject of the main differences are:First, because the random phenomena statistical regularity is a collective rule, must to present in a large number of similar random phenomena, therefore, observation, experiment, research is the cornerstone of the subject research methods of probability and statistics. But, as a branch of mathematics, it still has the definition of this discipline, axioms, theorems, the definitions and axioms, theorems are derived from the random rule of nature, but these definitions and axioms, theorems is certain, there is no randomness.Second, in the study of probability statistics, using the "by part concluded all" methods of statistical inference. This is because it the object of the research - the range of random phenomenon is very big, at the time of experiment, observation, not all may be unnecessary. But by this part of the data obtained from some conclusions, concluded that the reliability of the conclusion to all the scope.Third, the randomness of the random phenomenon, refers to the experiment, investigation before speaking. After the real results for each test, it can only get the results of the uncertainty of a certain result. When we study this phenomenon, it should be noted before the test can find itself inherent law of this phenomenon.The content of the theory of probabilityProbability theory as a branch of mathematics, it studies the content general include the probability of random events, the regularity of statistical independence and deeper administrative levels.Probability is a quantitative index of the possibility of random events. In independent random events, if an event frequency in all events, in a larger range of stable around a fixed constant. You can think the probability of the incident to the constant. For any event probability value must be between 0 and 1.There is a certain type of random events, it has two characteristics: first, only a finite number of possible results; Second, the results the possibility of the same. Have the characteristics of the two random phenomenon called"classical subscheme".In the objective world, there are a large number of random phenomena, the result of a random phenomenon poses a random event. If the variable is used to describe each random phenomenon as a result, is known as random variables.Random variable has a finite and the infinite, and according to the variable values is usually divided into discrete random variables and the discrete random variable. List all possible values can be according to certain order, such a random variable is called a discrete random variable; If possible values with an interval, unable to make the order list, the random variable is called a discrete random variable.The content of the mathematical statisticsIncluding sampling, optimum line problem of mathematical statistics, hypothesis testing, analysis of variance, correlation analysis, etc. Sampling inspection is to pair through sample investigation, to infer the overall situation. Exactly how much sampling, this is a very important problem, therefore, is produced in the sampling inspection "small sample theory", this is in the case of the sample is small, the analysis judgment theory.Also called curve fitting and optimal line problem. Some problems need to be according to the experience data to find a theoretical distribution curve, so that the whole problem get understanding. But according to what principles and theoretical curve? How to compare out of several different curve in the same issue? Selecting good curve, is how to determine their error? ...... Is belong to the scope of the optimum line issues of mathematical statistics.Hypothesis testing is only at the time of inspection products with mathematical statistical method, first make a hypothesis, according to the result of sampling in reliable to a certain extent, to judge the null hypothesis.Also called deviation analysis, variance analysis is to use the concept of variance to analyze by a handful of experiment can make the judgment.Due to the random phenomenon is abundant in human practical activities, probability and statistics with the development of modern industry and agriculture, modern science and technology and continuous development, which formed many important branch. Such as stochastic process, information theory, experimental design, limit theory, multivariate analysis, etc.译文：概率论和数理统计简介概率论与数理统计是对随机现象的统计规律进行演绎和归纳的科学，从数量侧面研究随机现象的统计规律性的基础数学学科，概率论与数理统计又可分为概率论和数理统计两个分支。

(完整word版)概率论与数理统计(英文)

3. Random Variables3.1 Definition of Random VariablesIn engineering or scientific problems, we are not only interested in the probability of events, but also interested in some variables depending on sample points. （定义在样本点上的变量）For example, we maybe interested in the life of bulbs produced by a certain company, or the weight of cows in a certain farm, etc. These ideas lead to the definition of random variables.1. random variable definitionHere are some examples.Example 3.1.1 A fair die is tossed. The number X shown is a random variable, it takes values in the set {1,2,6}.Example 3.1.2The life t of a bulb selected at random from bulbs produced by company A is a random variable, it takes values in the interval (0,) .Since the outcomes of a random experiment can not be predicted in advance, the exact value of a random variable can not be predicted before the experiment, we can only discuss the probability that it takes somevalue or the values in some subset of R.2. Distribution function Definition3.1.2 Let X be a random variable on the sample space S . Then the function()()F X P X x =≤. R x ∈is called the distribution function of XNote The distribution function ()F X is defined on real numbers, not on sample space.Example 3.1.3 Let X be the number we get from tossing a fair die. Then the distribution function of X is (Figure 3.1.1)0,1;(),1,1,2,,5;61, 6.if x n F x if n x n n if x <⎧⎪⎪=≤<+=⎨⎪≥⎪⎩Figure 3.1.1 The distribution function in Example 3.1.3 3. PropertiesThe distribution function ()F x of a random variable X has the following properties :(1) ()F x is non-decreasing.SolutionBy definition,1(2000)(2000)10.6321P X F e -≤==-=.(10003000)(3000)(1000)P X P X P X <≤=≤-≤1.50.5(3000)(1000)(1)(1)0.3834F F e e --=-=---= Question ： What are the probabilities (2000)P X < and (2000)P X =? SolutionLet 1X be the total number shown, then the events 1{}X k = contains 1k - sample points, 2,3,4,5k =. Thus11()36k P X k -==, 2,3,4,5k = And512{1}{}k X X k ==-==so 525(1)()18k P X P X k ==-===∑ 13(1)1(1)18P X P X ==-=-=Thus0,1;5()(),11;181, 1.x F x P X x x x <-⎧⎪⎪=≤=-≤<⎨⎪≥⎪⎩Figure 3.1.2 The distribution function in Example 3.1.5The distribution function of random variables is a connection between probability and calculus. By means of distribution function, the main tools in calculus, such as series, integrals are used to solve probability and statistics problems.3.2 Discrete Random Variables 离散型随机变量In this book, we study two kinds of random variables. ,,}n aAssume a discrete random variable X takes values from the set 12{,,,}n X a a a =. Let()n n P X a p ==,1,2,.n = (3.2.1) Then we have 0n p ≥, 1,2,,n = 1n n p=∑.the probability distribution of the discrete random variable X (概率分布)注意随机变量X 的分布所满足的条件(1) P i ≥0(2) P 1+P 2+…+P n =1离散型分布函数And the distribution function of X is given by()()n n a xF x P X x p ≤=≤=∑ (3.2.2)Solutionn=3, p=1/2X p r01/813/823/831/8two-point distribution(两点分布)某学生参加考试得5分的概率是p, X表示他首次得5分的考试次数，求X的分布。

概率论与数理统计(英文) 第三章

3. Random Variables3.1 Definition of Random VariablesIn engineering or scientific problems, we are not only interested in the probability of events, but also interested in some variables depending on sample points. （定义在样本点上的变量）For example, we maybe interested in the life of bulbs produced by a certain company, or the weight of cows in a certain farm, etc. These ideas lead to the definition of random variables.1. random variable definitionHere are some examples.Example 3.1.1 A fair die is tossed. The number X shown is a random variable, it takes values in the set {1,2,6}. Example 3.1.2The life t of a bulb selected at random from bulbs produced by company A is a random variable, it takes values in the interval (0,) .Since the outcomes of a random experiment can not be predicted in advance, the exact value of a random variable can not be predicted before the experiment, we can only discuss the probability that it takes somevalue or the values in some subset of R.2. Distribution functionNote The distribution function ()F X is defined on real numbers, not on sample space.Example 3.1.3 Let X be the number weget from tossing a fair die. Then the distribution function of X is (Figure 3.1.1)0,1;(),1,1,2,,5;61, 6.if x n F x if n x n n if x <⎧⎪⎪=≤<+=⎨⎪≥⎪⎩Figure 3.1.1 The distribution function in Example 3.1.3 3. PropertiesThe distribution function ()F x of a random variable X has the following properties :SolutionBy definition,1(2000)(2000)10.6321P X F e -≤==-=.(10003000)(3000)(1000)P X P X P X <≤=≤-≤1.50.5(3000)(1000)(1)(1)0.3834F F e e --=-=---= Question ： What are the probabilities (2000)P X < and (2000)P X =? SolutionLet 1X be the total number shown, then the events 1{}X k = contains1k - sample points, 2,3,4,5k =. Thus11()36k P X k -==, 2,3,4,5k = And 512{1}{}k X X k ==-==so525(1)()18k P X P X k ==-===∑ 13(1)1(1)18P X P X ==-=-=Thus0,1;5()(),11;181, 1.x F x P Xx x x <-⎧⎪⎪=≤=-≤<⎨⎪≥⎪⎩Figure 3.1.2 The distribution function in Example 3.1.53.2 Discrete Random Variables 离散型随机变量In this book, we study two kinds of random variables.Assume a discrete random variable X takes values from the set 12{,,,}n X a a a = . Let()n n P X a p ==,1,2,.n = (3.2.1) Then we have 0n p ≥, 1,2,,n = 1n n p=∑.the probability distribution of the discrete random variable X (概率分布)注意随机变量X 的分布所满足的条件(1) P i ≥0(2) P 1+P 2+…+P n =1离散型分布函数And the distribution function of X is given by()()n n a x F x P X x p ≤=≤=∑ (3.2.2)Solutionn=3, p=1/2X p r01/813/823/831/8two-point distribution(两点分布)某学生参加考试得5分的概率是p, X表示他首次得5分的考试次数，求X的分布。

(完整版)概率论与数理统计英文版第三章

Chapter 3. Random Variables and Probability Distribution1.Concept of a Random VariableExample: three electronic components are testedsample space (N: non defective, D: defective)S =｛NNN, NND, NDN, DNN, NDD, DND, DDN, DDD｝allocate a numerical description of each outcomeconcerned with the number of defectiveseach point in the sample space will be assigned a numerical value of 0, 1, 2, or 3.random variable X: the number of defective items, a random quantityrandom variableDefinition 3.1A random variable is a function that associates a real number with each element in the sample space.X: a random variablex : one of its valuesEach possible value of X represents an event that is a subset of the sample spaceelectronic component test:E =｛DDN, DND, NDD｝=｛X = 2｝.Example 3.1Two balls are drawn in succession without replacement from an urn containing 4 red balls and 3 black balls. Y is the number of red balls. The possible outcomes and the values y of the random variable Y ?Example 3.2 A stockroom clerk returns three safety helmets at random to three steel mill employees who had previously checked them. If Smith, Jones, and Brown, in that order, receive one of the three hats, list the sample points for the possible orders of returning the helmets,and find the value m of the random variable M that represents the number of correct matches.The sample space contains a finite number of elements in Example 3.1 and 3.2.another example: a die is thrown until a 5 occurs,F: the occurrence of a 5N: the nonoccurrence of a 5obtain a sample space with an unending sequence of elementsS =｛F, NF, NNF, NNNF, . . .｝the number of elements can be equated to the number of whole numbers; can be countedDefinition 3.2 If a sample space contains a finite number of possibilities or an unending sequence with as many elements as there are whole numbers, it is called a discrete sample space.The outcomes of some statistical experiments may be neither finite nor countable.example: measure the distances that a certain make of automobile will travel over a prescribed test course on 5 liters of gasolinedistance: a variable measured to any degree of accuracywe have infinite number of possible distances in the sample space, cannot be equated to the number of whole numbers.Definition 3.3If a sample space contains an infinite number of possibilities equal to the number of points on a line segment, it is called a continuous sample spaceA random variable is called a discrete random variable if its set of possible outcomes is countable.Y in Example 3.1 and M in Example 3.2 are discrete random variables.When a random variable can take on values on a continuous scale, it is called a continuous random variable.The measured distance that a certain make of automobile will travel over a test course on 5 liters of gasoline is a continuous random variable.continuous random variables represent measured data:all possible heights, weights, temperatures, distance, or life periods.discrete random variables represent count data: the number of defectives in a sample of k items, or the number of highway fatalities per year in a given state.2. Discrete Probability DistributionA discrete random variable assumes each of its values with a certain probabilityassume equal weights for the elements in Example 3.2, what's the probability that no employee gets back his right helmet.The probability that M assumed the value zero is 1/3.The possible values m of M and their probabilities are0 1 31/3 1/2 1/6Probability Mass FunctionIt is convenient to represent all the probabilities of a random variable X by a formula.write p(x) = P (X = x)The set of ordered pairs (x, p(x)) is called the probability function or probability distribution of the discrete random variable X.Definition 3.4The set of ordered pairs (x, p(x)) is a probability function, probability mass function, or probability distribution of the discrete random variable X if, for each possible outcome xExample 3.3 A shipment of 8 similar microcomputers to a retail outlet contains 3 that are defective. If a school makes a random purchase of 2 of these computers, find the probability distribution for the number of defectives.SolutionX: the possible numbers of defective computersx can be any of the numbers 0, 1, and 2.Cumulative FunctionThere are many problem where we may wish to compute the probability that the observed value of a random variable X will be less than or equal to some real number x.Definition 3.5The cumulative distribution F (x) of a discrete random variable X with probability distribution p(x) isFor the random variable M, the number of correct matches in Example 3.2, we haveThe cumulative distribution of M isRemark. the cumulative distribution is defined not only for the values assumed by given random variable but for all real numbers.Example 3.5 The probability distribution of X isFind the cumulative distribution of the random variable X.Certain probability distribution are applicable to more than one physical situation.The probability distribution of Example 3.5 can apply to different experimental situations.Example 1: the distribution of Y , the number of heads when a coin is tossed 4 timesExample 2: the distribution of W , the number of read cards that occur when 4 cards are drawn at random from a deck in succession with each card replaced and the deck shuffled before the next drawing.graphsIt is helpful to look at a probability distribution in graphic form.bar chart;histogram;cumulative distribution.3.Continuous Probability DistributionContinuous Probability distributionA continuous random variable has a probability of zero of assuming exactly any of its values. Consequently, its probability distribution cannot be given in tabular form.Example: the heights of all people over 21 years of age (random variable)Between 163.5 and 164.5 centimeters, or even 163.99 and 164.01 centimeters, there are an infinite number of heights, one of which is 164 centimeters.The probability of selecting a person at random who is exactly 164 centimeters tall and not one of the infinitely large set of heights so close to 164 centimeters is remote.We assign a probability of zero to a point, but this is not the case for an interval. We will deal with an interval rather than a point value, such as P (a < X < b), P (W≥c).P (a≤X≤b) = P (a < X≤b) = P (a≤X < b) = P (a < X < b)where X is continuous. It does not matter whether we include an endpoint of the interval or not. This is not true when X is discrete.Although the probability distribution of a continuous random variable cannot be presented in tabular form, it can be stated as a formula.refer to histogramDefinition 3.6 The function f(x) is a probability density function for the continuous random variable X, defined over the set of real numbers R, ifExample 3.6 Suppose that the error in the reaction temperature, in oC, for a controlled laboratory experiment is a continuous random variable X having the probability density function(a) Verify condition 2 of Definition 3.6.(b) Find P (0 < X≤1).Solution: . . . . . . P (0 < X≤1) = 1/9.Definition 3.7The cumulative distribution F (x) of a continuous random variable X with density function f(x) isimmediate consequence:Example 3.7 For the density function of Example 3.6 findF (x), and use it to evaluate P (0 < x≤1).4. Joint Probability Distributionsthe preceding sections: one-dimensional sample spaces and a single random variablesituations: desirable to record the simultaneous outcomes of several random variables.Joint Probability DistributionExamples1. we might measure the amount of precipitate P and volume V of gas released from a controlled chemical experiment; we get a two-dimensional sample space consisting of the outcomes (p, v).2. In a study to determine the likelihood of success in college, based on high school data, one might use a three-dimensional sample space and record for each individual his or her aptitude test score, high school rank in class, and grade-point average at the end of the freshman year in college.X and Y are two discrete random variables, the joint probability distribution of X and Y isp (x, y) = P (X = x, Y = y)the values p(x, y) give the probability that outcomes x and y occur at the same time.Definition 3.8 The function p(x, y) is a joint probability distribution or probability mass function of the discrete random variables X and Y ifExample 3.8Two refills for a ballpoint pen are selected at random from a box that contains 3 blue refills,2 red refills, and 3 green refills. If X is the number of blue refills and Y is the number of red refills selected, find(a) the joint probability function p(x, y)(b) P [(X, Y )∈A] where A is the region｛(x, y)|x + y≤1｝Solutionthe possible pairs of values (x, y) are (0, 0), (0, 1), (1, 0), (1, 1), (0, 2), and (2, 0).p (x, y) represents the probability that x blue and y red refills are selected.present the results in Table 3.1Definition 3.9The function f(x, y) is a joint density function of the continuous random variables X and Y ifWhen X and Y are continuous random variables, the joint density function f(x, y) is a surface lying above the xy plane.P [(X, Y )∈A], where A is any region in the xy plane, is equal to the volume of the right cylinder bounded by the base A and the surface.Example 3.9 Suppose that the joint density function is(b) P [(X, Y )∈A]= 13/160marginal distributionp (x, y): the joint probability distribution of the discrete random variables X and Ythe probability distribution p X(x) of X alone is obtained by summing p(x, y) over the values of Y .Similarly, the probability distribution p Y (y) of Y alone is obtained by summing p(x, y) over the values of X.pX (x) and p Y (y): marginal distributions of X and YWhen X and Y are continuous random variables, summations are replaced by integrals.Definition 3.10The marginal distribution of X alone and of Y alone areExample 3.10 Show that the column and row totals of Table3.1 give the marginal distribution of X alone and of Y alone.Example 3.11 Find marginal probability density functionsf X (x) and fy(y)for the joint density function of Example 3.9.The marginal distribution pX (x) [or fX(x)] and px(y) [or fy(y)] are indeed the probability distribution of the individual variableX and Y , respectively.How to verify?The conditions of Definition 3.4 [or Definition 3.6] are satisfied.Conditional distributionrecall the definition of conditional probability:X and Y are discrete random variables, we haveThe value x of the random variable represent an event that is a subset of the sample space.Definition 3.11Let X and Y be two discrete random variables. The conditional probability mass function of the random variable Y , given that X = x, isSimilarly, the conditional probability mass function of the random variable X, given that Y = y, isDefinition 3.11'Let X and Y be two continuous random variables. The conditional probability density function of the random variable Y , given that X = x, isSimilarly, the conditional probability density function of the random variable X, given that Y = y, isRemark:The function f(x, y)/fX (x) is strictly a function of y with x fixed, the function f(x, y)/fy(y) is strictly a function of x with yfixed, both satisfy all the conditions of a probability distribution.How to find the probability that the random variable X falls between a and b when it is known that Y = yExample 3.12 Referring to Example 3.8, find the conditional distribution of X, given that Y = 1, and use it to determineP (X = 0|Y = 1).Example 3.13The joint density for the random variables (X, Y ) where X is the unit temperature change and Y is the proportion of spectrum shift that a certain atomic particle produces is(a)Find the marginal densities fX (x), fy(y), and the conditional density f Y |X (y|x)(b)Find the probability that the spectrum shifts more than half of the total observations, given the temperature is increased to0 .25 unit.(a)(b)Example 3.14 Given the joint density function(a)(b)statistical independenceevents A and B are independent, ifP (B∩A) = P (A)P (B).discrete random variables X and Y are independent, ifP (X = x, Y = y) = P (X = x)P (Y = y)for all (x, y) within their range.The value x of the random variable represent an event that is a subset of the sample space.Definition 3.12 Let X and Y be two discrete random variables, with joint probability distribution p(x, y) and marginaldistributions pX (x)and pY(y), respectively. The random variables X and Y are said to be statistically independent if and onlyifp (x,y) = pX (x)pY(y) for all (x, y) within their range.Definition 3.12' Let X and Y be two continuous random variables, with joint probability distribution f(x, y) and marginaldistributions fX (x) and fY(y), respectively. The random variables X and Y are said to be statisticallyindependent if and only iff (x, y) =fX (x)fY(y) for all (x, y) within their range.The continuous random variables of Example 3.14 are statistically independent. However, that is not the case for the Example 3.13.For discrete variables, requires more thorough investigation. If you find any point (x, y) for which p(x, y) is defined such that≠pX (x)pY(y), the discrete variables X and Y are not statistically independent.p(x, y)Example 3.15Show that the random variables of Example 3.8 are notstatistically independent.the case of n random variablesjoint marginal distributions of two r.v. X1 and X2Definition 3.13Let x1, x2,…, x n be n discrete random variables, with joint probability distribution p(x1, x2,… , x n)and marginal distributions p X1 (x1), p X2 (x2),…, p Xn (x n),respectively.The random variables x1, x2,…, x n are mutually statistically independent,thenfor all (x1, x2,… , x n) within their range.Definition 3.13' Let x 1, x2,…, x n be n continuous randomvariables, with joint probability distribution f(x1, x2,… , x n)and marginal distributions f X1 (x1),f X2 (x2),…, f Xn (x n)respectively. The random variables x1, x2,…, x n are mutually statistically independent, thenfor all(x1, x2,… , x n)within their range.Example 3.16 Suppose that the shelf life , in years, of a certain perishable food product packaged in cardboard containers is a random variable whose probability density function is given by。

概率论与数理统计(英文) 第七章

7. Estimation Problems7.1 Point Estimate 点估计1. conceptEstimation is one of the most important problems in statistics. Why ----the distribution depends on the parameter.Poisson distribution ),(λk PNormal distribution ),(2σμNProblems of estimation include(1) determining parameters of a population from the random samples, estimating the mean or variance of a population by sampling,(2) how to judge if such estimation is a “good” one.Point Estimation ----- use the value of a statistic to estimate a parameterFor example, if we use X to estimate the parameter λ of a Poisson population, X is then the estimator of λ and x is a point estimate of the parameter λ.Point estimation is not unique.For example, we often use X orM----- mean.2. Judge(1) Unbiased 无偏(2) asymptotically unbiased 一致(3) minimum variance unbiased estimator. 最有效无偏估计Proof.Since for a Poisson we know thatλ(PE=)and by Theorem 6.3.1E(PE=,X)()we conclude thatλE=(X)which shows that X is an unbiased estimator of λ. □Proof. Suppose μ is the mean of the population, by Theorem 6.4.1（P100）, we have thatωωμωωωωωω≠+=+-=⋅==⎰⎰∞--∞--∞--1)()()()(dx e xe dx e x X E x x x . Therefore, X is a biased estimator of ω.均值的点估计 First of all, let us discuss the situation of using point estimations. X is usually the most common choice of the point estimator of the mean. By Theorem 6.4.1, we can easily obtain the following theorem.Example 7.1.3 Let 2ˆSbe the variance of a random sample of size n from a random variable X which has a finite variance 2σ. Show that 2ˆSis an asymptotically unbiased estimator of 2σ.Proof. By Theorem 6.3.1,)()()()(11)ˆ(222122122X E X E X E X E n X X n E S E n i i n i i -=-=⎪⎭⎫ ⎝⎛-=∑∑==. Since []22)()()(X E X E X D -= and []22)()()(X E X E X D -=, we have []{}[]{}).()()()()()()ˆ(222X D X D X E X D X E X D S E -=+-+= Here,n X D n X D n X D n X n D X D n i n i i n i i 212121)(1)(1)(11)(σ====⎪⎭⎫ ⎝⎛=∑∑∑===. Thus, we get that22221)ˆ(σσσnn n S E -=-=. We can see that 2ˆSis a biased estimator of 2σ. But when n is large, 2221lim )ˆ(lim σσ=-=∞→∞→nn S E n n which leads to the conclusion that 2ˆSis an asymptotically unbiased estimator of 2σ.堂上练习Show that 22)(σ=S E ， E ∑=--=n i i X X n S 122)(11=?As mentioned before, there exists more than one unbiased estimators for the mean μ. Therefore, it is meaningful to study, under those popular distributions, whether X is also the minimum variance unbiased estimator. Here, an important tool is Cramer-Rao inequality .(最小方差无偏估计)Cramer-Rao inequality⎥⎥⎦⎤⎢⎢⎣⎡⎪⎭⎫ ⎝⎛∂∂≥2)(ln 1)ˆ(θθX f nE Dwhere )(x fis the population density function and n is the sample size.minimum variance unbiased estimator of μ.Proof . In this example, the population density function is22121)(⎪⎭⎫ ⎝⎛--=σμπσx e x fwhich leads to()⎪⎭⎫ ⎝⎛-=⎥⎥⎦⎤⎢⎢⎣⎡⎪⎭⎫ ⎝⎛---∂∂=∂∂σμσσμπσμμx x x f 1212ln )(ln 2. Therefore,222222111)(ln σσμσσμσμ=⎥⎥⎦⎤⎢⎢⎣⎡⎪⎭⎫ ⎝⎛-=⎥⎥⎦⎤⎢⎢⎣⎡⎪⎭⎫ ⎝⎛-=⎥⎥⎦⎤⎢⎢⎣⎡⎪⎪⎭⎫ ⎝⎛∂∂X E X E X f E since )1,0(~N X σμ-. Thus, )(11)(ln 1222X D n n X f nE ==⋅=⎥⎥⎦⎤⎢⎢⎣⎡⎪⎭⎫ ⎝⎛∂∂σσθ. According to Theorem 7.1.1, we conclude that X is a minimum variance unbiased estimator of μ.3. Method of Moments 矩估计的方法Example 7.1.7求总体的数学期望和方差的矩估计。

《概率论与数理统计》教学大纲课程名称：概率论与数理统计英文名称

《概率论与数理统计》教学大纲课程名称：概率论与数理统计英文名称：Probability Theory and Mathematical Statitics课程编号：09420003学时数及学分：54学时 3学分教材名称及作者：《概率论与数理统计》（第三版）, 盛骤、谢式干、潘承毅编出版社、出版时间：高等教育出版社，2001年本大纲主笔人：邓娜一、课程的目的、要求和任务概率统计是一门重要的理论性基础课，是研究随机现象统计规律性的数学学科，本课程的任务是使学生掌握概率论与数理统计的基本概念，了解它的基本理论和方法，从而使学生初步掌握处理随机现象的基本思想和方法，培养学生运用概率统计方法分析和解决、处理实际不确定问题的基本技能和基本素质。

通过本课程的学习，要使学生初步理解和掌握概率统计的基本概念和基本方法，了解其基本理论，学习和训练运用概率统计的思想方法观察事物、分析事物以及培养学生用概率统计方法解决实际问题的初步能力。

概率统计的理论和方法的应用是非常广泛的，几乎遍及所有科学技术领域，工农业生产和国民经济的各个部门，例如使用概率统计方法可以进行气象预报，水文预报以及地震预报，产品的抽样检验，在研究新产品时，为寻求最佳生产方案可以进行试验设计和数据处理，在可靠性工程中，使用概率统计方法可以给出元件或系统的使用可靠性以及平均寿命的估计，在自动控制中，可以通过建立数学模型以便通过计算机控制工业生产，在通讯工程中可用以提高抗干扰和分辨率等。

所以我院各专业学习概率统计是非常必要的，它也是学习专业课的基础。

二、大纲的基本内容及学时分配本课程的教学要求分为三个层次。

凡属较高要求的内容，必须使学生深入理解、牢固掌握、熟练应用。

其中，概念、理论用“理解”一词表述，方法、运算用“熟练掌握”一词表述。

在教学要求上一般的内容中，概念、理论用“了解”一词表述，方法、运算用“掌握”表述。

对于在教学上要求低于前者的内容中，概念、理论用“会”一词表述，方法、运算用“知道”表述（一）随机事件及其概率1、理解随机实验、随机事件、必然事件、不可能事件等概念。

概率论与数理统计(英文)

连续型随机变量
1.定义
定义4.1.1函数f(x)定义在上，且满足下面两个条件
(i) ;
(ii)f(x)在上可积且 .
那么f(x)就叫概率密度函数.
定义4.1.2设f（x）是一个概率密度函数。如果x是一个具有分布函数的随机变量且
(4.1.1)
那么x称为一个具有连续随机变量的密度函数f（x）。在这种情况下
定义4.5.1一个连续变量X有一个参数的指数分布，其密度函数为
定理4.5.1连续型随机变量X具有指数分布的均值和方差由下式给出
If the continuous random variable x withprobability density function
Called X in the (a, b) obey uniform distribution
4. Exponential Distribution指数分布
Many random variables, such as the life of automotive parts, life of animals, time period between two calls arrives to an office, having a distribution called exponential distribution.
Continuous Random Variable
1.Definition
Definition 4.1.1
A functionf(x) defined on is called aprobability density function（概率密度函数）if:
(i) ;
(ii)f(x) is intergrable on and .

《概率论与数理统计》(全英语)教学大纲课程名称概率

《概率论与数理统计》(全英语)教学大纲课程名称:概率论与数理统计学时:48学时学分:2.5分先修课程:高等数学,线性代数开课院系:上海交通大学理学院数学系教材:华章统计学原版精品系列：概率统计（英文版·第4版）, [美]德格鲁特（Morris H.DeGroot），[美]舍维什（Mark J.Schervish）著Morris H.DeGroot ，Mark J.Schervish 编, 机械工业出版社, 2012教学参考:[1] M.N. DeGroot, M.J. Schervish, Probability and Statistics, 3rd ed. Boston, MA; London:Addison-Wesley, 2002[2] Jay.L. Devore, Probability and Statistics, 5th ed. Higher Education Press, 2010[3] H. Jeffreys, Theory of Probability, 3rd ed. Oxford: Oxford University Press, 1998[4] J.T. McClave, T. Sincich, A First Course in Statistics, 7th ed. Upper Saddle River, NJ: PrenticeHall; London: Prentice-Hall International, 2000[5] S.M. Ross, Introduction to Probability and Statistics for Engineers and Scientists,2nd ed. SanDiego, CA; London: Harcourt/Academic, 2000[6] V.K. Rothagi, S.M. Ehsanes, An Introduction to Probability and Statistics, 2nd ed.New York, Chichester: Wiley, 2001Probability and Statistics (English)Curriculum IntroductionCourse Title: Probability and Statistics (English)Total Hours: 48Credit: 2.5Pre-Course：Calculus, Linear AlgebraDepartment of giving course: Department of mathematics in Shanghai Jiaotong UniveristyTextbook:Probability and Statistics ( fourth edition), [美]德格鲁特（Morris H.DeGroot），[美]舍维什（Mark J.Schervish）著Morris H.DeGroot ，MarkJ.Schervish 编, 机械工业出版社, 2012Reference:[1] M.N. DeGroot, M.J. Schervish, Probability and Statistics, 3rd ed. Boston, MA; London: Addison-Wesley, 2002[2] Jay.L. Devore, Probability and Statistics, 5th ed. Higher Education Press, 2010[3] H. Jeffreys, Theory of Probability, 3rd ed. Oxford: Oxford University Press, 1998[4] J.T. McClave, T. Sincich, A First Course in Statistics, 7th ed. Upper Saddle River, NJ: Prentice Hall; London: Prentice-Hall International, 2000[5] S.M. Ross, Introduction to Probability and Statistics for Engineers and Scientists,2nd ed. San Diego, CA; London: Harcourt/Academic, 2000[6] V.K. Rothagi, S.M. Ehsanes, An Introduction to Probability and Statistics, 2nd ed. New York, Chichester: Wiley, 2001<<概率论与数理统计>>是一门从数量方面研究随机现象规律性的数学学科，它已广泛地应用于工农业生产和科学技术之中，并与其它数学分支互相渗透与结合。

概率论与数理统计(英文)第三章

3. Random Variables3.1 Definition of Random VariablesIn engineering or scientific problems, we are not only interested in the probability of events, but also interested in some variables depending on sample points. （定义在样本点上的变量）For example, we maybe interested in the life of bulbs produced by a certain company, or the weight of cows in a certain farm, etc. These ideas lead to the definition of random variables.1. random variable definitionHere are some examples.Example 3.1.1 A fair die is tossed. The number X shown is a random variable, it takes values in the set {1,2,6}.Example 3.1.2The life t of a bulb selected at random from bulbs produced by company A is a random variable, it takes values in the interval (0,) .Since the outcomes of a random experiment can not be predicted in advance, the exact value of a random variable can not be predicted before the experiment, we can only discuss the probability that it takes somevalue or the values in some subset of R.2. Distribution functionNote The distribution function ()F X is defined on real numbers, not on sample space.Example 3.1.3Let X be the number we get from tossing a fair die. Then the distribution function of X is (Figure 3.1.1)Figure 3.1.1 The distribution function in Example 3.1.3 3. PropertiesThe distribution function ()F x of a random variable X has the following properties:SolutionBy definition,1(2000)(2000)10.6321P X F e -≤==-=.Question ： What are the probabilities (2000)P X < and (2000)P X =?SolutionLet 1X be the total number shown, then the events 1{}X k = contains 1k - sample points, 2,3,4,5k =. Thus11()36k P X k -==, 2,3,4,5k = AndsoThus Figure 3.1.2 The distribution function in Example 3.1.53.2 Discrete Random Variables 离散型随机变量In this book, we study two kinds of random variables. ,,}n aAssume a discrete random variable X takes values from the set 12{,,,}n X a a a =. Let()n n P X a p ==,1,2,.n = (3.2.1) Then we have 0n p ≥, 1,2,,n = 1n n p=∑.the probability distribution of the discrete random variable X (概率分布)注意随机变量X 的分布所满足的条件(1) P i ≥0(2) P 1+P 2+…+P n =1离散型分布函数 And the distribution function of X is given by()()n n a xF x P X x p ≤=≤=∑ (3.2.2)Solutionn=3, p=1/2X p r01/813/823/831/8two-point distribution(两点分布)某学生参加考试得5分的概率是p, X表示他首次得5分的考试次数，求X的分布。

概率论与数理统计(英文) 第六章

（---24 | 25----）6. Fundamental Sampling Distributions and Data Descriptions抽样分布 6.1 Analysis of Data Mean 均值 Kx x x m K+++= 21mmm f f f f x f x f x m ++++++=212211,Median 中位数()⎪⎩⎪⎨⎧+=++2/)1(2/2121k k k d x x x MMean Deviation 平均差（均值离差）∑-=Kiim xKD M 1..方差()∑=-=Ki i m x K 1221σ标准差 ()∑=-=Ki i m x K 121σ 6．2 Random SamplingSampling is one of the most important concepts in the study of statistics. We need the fundamental ideas of populations and samples before studying particular statistical descriptions.population (总体)sample （样本）sampling （抽样）For example, in the study of the grade of Calculus course in some university of the year 2006, the grades of all the students who took this course constitute the population.a finite population（有限总体）infinite population（无限总体）This is a finite population. A example is the study of the length of newborns in China.The population is then the all possible lengths of the newborns in China, in the past, now or in the future. Such population is an infinite one.Since in many cases we are not able to investigate a whole population we are obliged to get conclusions regarding a population from its samples.sample （样本、子样）sampling(抽样)taking a sample: The process of performing an experiment to obtain a sample from the population is called sampling.sampling is done with replacement 有放回抽样sampling is done without replacement 无放回(有放回抽样，使样本点独立同分布)The purpose of the sampling is to find out something about the nature of the population.good sample——→good estimatesgood estimates concerning a population necessitate good sample.good sample-----random sample (随机样本)If ),,,(21n x x x f is the value of the joint distribution of such a set of random variables at ),,,(21n x x x , we can get that∏==ni i n x f x x x f 121)(),,,( (6.1.1)in which )(i x f is the population distribution at i x .6.3 Statistics 统计量We will discuss the definitionof the word statistic and someSample variance 子样方差Remark:The sample variance is sometimes defined as212)(11X X n S ni i --=∑= in some books.Here, X and 2S are both random variables. Suppose a set of values of observations of the random sample is n x x x ,,,21 , then the observation value of X and 2S are denoted as 观察值2121)(1ˆ,1x x n sx n x n i i ni i -==∑∑==.In the future, normally we use capital letters to represent random variables and use small letters to represent the observation values.顺序统计量With the definition of the order statistics, we are able to introduce some more terms that are also useful and important. 中位数Example 6.2.1 54321,,,,X X X X X is a random sample of size 5. If an observation of this sample yields the values 2, 5, 1, 4, 8. Then, we can get the value of statistics as the following.Sample mean : 4)84152(51=++++=xSample variance : 64)84152(51ˆ2222222=-++++=s)5()4()3()2()1(,,,,X X X X X : 1, 2, 4, 5, 8Sample median : 40=mSample range: 718=-=r □6.4 Sample Distributions 抽样分布It should be kept in mind that a statistic , being computed from samples, is a random variable .1．sampling distribution of the mean 均值的抽样分布Proof . First,μμ===⎪⎭⎫ ⎝⎛=∑∑∑===n i ni i n i i n X E n X n E X E 1111)(11)(.Second, since each pair of i X and j X , with j i ≠, are independent. We can get thatnn X D nX n D X D ni ni in i i 21221211)(1)1()(σσ∑∑∑=======. □It is customary to write )(X E as X μ and )(X D as 2X σ.Here, ()E X μ= is called the expectation of the mean .均值的期望nX σσ=is called the standard error of the mean . 均值的标准差This formula shows that the standard deviation of the distribution of X decreases when n , the sample size, is increased. It means that when n becomes larger, we can expect that the value of X to be closer to μ. 方差越小表示子样均值越靠近总体均值总体服从正态分布X ~),(2σμN ，则 ),(~2n N X σμ，总体分布未知，但n 够大，则X ~ N(μ, σ2/n)（子样均值也近似服从正态）Example 8-1 find P{X >68.9} Solution.Since2~(,)x X X N μσ,68.20x μμ==， 25.010050.2===n x σσ2~()X Nσμ~(0,1)Nso P{X>68.9}=68.968.20}0.25P->=P{Z > 2.8}=1－P{Z < 2.8}=0.5 – 0.4974=0.0026Distribution of the sample standard Deviation （子样标准差的分布）the population with mean=mstandard deviation=σ the sample standard deviation21)(1ˆX X n Sni i -=∑=for large nσμ=S ˆ,nS 2ˆσσ=此处并没有给出具体的分布，但告诉我们，当n 足够大时， *****子样标准差的均值 s ES μσ==, 和子样标准差的标准差S σ==Homework 6.1 6.6 6.8堂上练习设总体服从正态分布N(12,4) ,今抽取容量为5的样本X 1,X 2,X 3,X 4,X 5，试问：（1）样本均值 X 大于13的概率是多少？（2）样本均值X 的数学期望E X 、方差D X 、子样方差的数学期望是多少？（3）如果（1，0，3，1，2）是样本的一个观察值，它的样本平均值和样本方差等于多少？数据处理（找数据的特征、规律） 1．直方图 histogramExample 6.3.2 We are interested in the distribution of peoples ’ age in some city. In our hand, we have a sample of the ages of 50 people who were randomly picked and the data are listed below:35 23 48 21 15 36 12 3 14 54 21 43 92 15 32 5 71 55 62 24 64 78 50 11 9 43 28 25 34 15 22 30 51 88 16 75 22 27 41 33 45 61 35 28 43 59 38 90 70 6 Find the frequency table and draw the histogram. Solution .We find that the minimum sample value in this example is 3)1(=x and the maximum sample value is 92)50(=x . So, we choose the range of the frequency table as ]100,0[ and divided it into 10 classes : [0, 10), [10, 20), [20, 30), [30, 40), [40, 50), [50, 60), [60, 70), [70, 80), [80, 90), [90, 100].Then, we count the number of sample values in each class and get the following frequency table.Table 6.3.1At last, using the horizontal axis to represent the classes and the vertical axis to represent the frequency, we can draw the histogram below. □In summery, for a random sample of size n , the process of drawing a histogram is as following: 方法步骤(1) Find the minimum sample value )1(x and maximum sample value)(n x . Decide the number of classes m (usually between 5 and 20) and thendecide a fitting range ],[0m a a of the frequency table so that 0a is a little less than )1(x and m a is a little bigger than )(n x .(2) Insert 1-m points into ],[0m a a : m m a a a a a <<<<<-1210 so that the range ],[0m a a is divided into m equal length classes: ),[10a a ,),[21a a , ,),[12--m m a a , ],[1m m a a -. The length of each class isma a m 0-. (3) Count for each class how many sample values are inside this class, and then get the frequency table.(4) Using the horizontal axis to represent the classes and the vertical axis to represent the frequency, draw the histogram at last.2．带频数的样本均值和样本方差For Table 6.3.1 , How to find the sample mean and sample variance?Table 6.3.1Example 6.3.2Find the sample mean and the sample variance if we have a random sample that is given in Table 6.3.1. Solution . By Definition 6.3.5, we get that()4.39295185475365555645835102571545501=⨯+⨯+⨯+⨯+⨯+⨯+⨯+⨯+⨯+⨯=x []64.5562)4.3995(1)4.3985(4)4.3975(3)4.3965(5)4.3955(6)4.3945(8)4.3935(10)4.3925(7)4.3915(4)4.395(501ˆ22222222222=⨯-+⨯-+⨯-+⨯-+⨯-+⨯-+⨯-+⨯-+⨯-+⨯-=s*****==mi i f N 16.5 Chi-square Distributions1．gamma distribution ,),(~βαΓXgamma function⎰∞--=Γ01)(dy e y y αα for 0>α.Gamma function has a few useful properties, such the recursion formula)1()1()(-Γ-=Γααα.Also, when n is a positive integer, we have)!1()(-=Γn n .At last, an important special value is π=Γ)21( Properties 性质（有可加性）The gamma densities with several special values of α and β are shown in Figure 6.5.1. The readers can get some idea about the shape of the gamma distribution.12340.51.01.52.02.5xf(x)α=1/2, β=1α=2, β=1/2α=10, β=1/5Figure 6.4.1: Graphs of gamma distribution),(~βαΓX2．特例（1）1=α and λβ=，),1(),(λβαΓ=Γ----- exponential distributionthe probability density of ),1(λΓ is⎪⎩⎪⎨⎧≤>=-0for 00for 1)(x x e x f xλλ（2） 2να= and 2=β，)2,2(),(νβαΓ=Γ---- chi-square distributionChi-square distribution)(2~νχX 卡方分布the parameter ν ----- degree of freedom （自由度）. We will write)(2~νχX ifXis a random variable which follows achi-square distribution with the degree of freedom ν.Corollary 6.5.1 The mean and the variance of the chi-square distribution)(2~νχX is given byν=)(X E and ν2)(=X D .The chi-square distribution is closely related to the normal distribution and has many important applications in statistics. Let us list several meaningful properties below.卡方分布与正态分布有密切联系，在统计学中有重要的应用3．重要结论Proof . We omit the proof of part (i) since it is beyond the scope of this book. Let us show only the part (ii).In order to study the distribution of22σnS , we need the identity22112)()()(μμ-+-=-∑∑==X n X X Xni i ni i(6.5.5)In fact, the left hand side of (6.4.5) is21221121221222)2()(μμμμμμμn X n X n X X X X Xni i ni i ni i ni i i ni i+-=+-=+-=-∑∑∑∑∑=====and the right hand side of (6.5.5) is()μμμμμμμn X n X n X n X n X n X X n X n X n X n X X X X n X X ni i ni i ni i n i i ni i +-=+-+=+-+⎪⎭⎫ ⎝⎛+-=-+-∑∑∑∑∑=====222222)()(122212222112221－which proves (6.4.5). By the definition of 2S in (6.3.2), we have212)(∑=-=ni i X X nS . Substitute it into (6.4.5) and then divide both sides of(6.5.5) by 2σ to get22212/⎪⎪⎭⎫ ⎝⎛-+=⎪⎭⎫ ⎝⎛-∑=n X nS X ni i σμσσμ (6.5.6)We know that)1,0(~N X i σμ- and from Theorem 6.5.4, weconclude that )(212~n ni i X χσμ∑=⎪⎭⎫⎝⎛-. At the same time, by Theorem 6.4.1and Theorem 6.5.3, we get that )1(22~/χσμ⎪⎪⎭⎫⎝⎛-n X . Therefore, thanks Theorem 6.4.6, we have that)1(222~-n nS χσ.Let X be a random variable whose distribution is )5(2χ. Find thevalue a such that 05.0)(=≥a X P6.6 Student’s Distributions (t -Distribution)学生分布/****Consider the case that a random sample of size n from a normal population with the mean μ and the variance 2σ. In Corollary 6.3.1, we know that the random variable X is also a normal distribution),(2nN σμ. Furthermore,)1,0(~/N nX σμ-. (6.5.1)Of course this is an important conclusion in statistics. However, in application, the population standard deviation σ is usually unknown. Therefore, people start to seek a replacement of σ with an estimate. Naturally, the sample standard deviation S seems to be a good choice. Since in the later material, we will know that nn S E σ=-)1(, wereplacenσin (6.5.1) by1-n Sand it comes the problem: What is the exact distribution of 1/--n S X μ for a random sample from a normalpopulation?This problem was originally studied by W.S. Gosset, who wrote under the pen name “student” because the company where he worked, Guiness’ Brewery in Dublin, did not allow publication by the employees. Gosset found that the quantities resulting from this substitution no longer follows normal distribution, instead, it satisfied a different type of distribution which has been called Student’s t -distribution since then. Let us first introduce a more general situation which leads to the formal definition of t-distribution. ****/1．Student ’s distributionWe omit the detailed proof which is above the requirement of this book. If a random distribution T has the t-distribution with ν degrees of freedom, we will write )(~νt T for short.-3-2-11230.10.20.30.4xf(x)←ν=1←ν=5←Standard normal distributionFigure 6.6.1Figure 6.6.1 includes the graphs of standard normal distribution, t-distributions with 1 and 5 degrees of freedom. We can see that the curves of t-distributions resemble in general shape the normal distribution. Also, as ν increases, the t-distribution will get closer to the normal distribution. In fact, the standard normal distribution can be considered to be the limiting case for the )(νt distributions, that is, ∞=ν.With the help of Theorem 6.6.1, we are now able to obtain the distribution of1/--n S X μ which is the problem raised in the beginning ofthis section. It is treated in the following theorem.2. 应用Proof. LetnX Z /σμ-=and 22ˆσSn Y =.By the Corollary 6.4.1 and Theorem 6.5.7, we get that )1,0(~N Z and)1(2~-n Y χ, also, Z and Y are independent. Thus1/ˆ)1(ˆ/)1/(22--=--=-n SX n Sn n X n Y Z μσσμand by Theorem 6.5.1, we conclude that)1(~1/ˆ---n t n SX μ. □Let Y be a t -distribution with degrees of freedom n , find the value a such that (a) 10.0)(=>a Y P , when 5=n ; (b)10.0)(=≤a Y P , when 5=n .Homework 6.9, 6.14, 6.17αα=>)}({n t t P6.6 F-DistributionsIn this section, we will study another important distribution in practical applications of statistics, the F-distribution. It helps us to deal with the comparison of the variability of two samples. In order to obtain the F-distribution, we consider the ratio of two independent chi-square random variables, each divided by its own degrees of freedom. The distribution of this radio is presented in the following theorem.Theorem 6.6.1 is often applied in the cases when we are interested in comparing the variances of two normal populations or two samples from normal populations. For example, we may need to estimate the ratio2221σσ. Or, in many cases we need to check whether 21σσ=.We already know, from Theorem 6.4.7, that if 21S is the variance of a random sample of size 1n from a normal population with variance21σ, and 22S is the variance of a random sample of size 2n from anormal population with variance 22σ, thenSuppose that these two samples are independent random samples , which means that all the 21n n + random variables in these two random samples are independent. Then these two chi-square random variables are also independent. Replacing X and Y in Theorem 6.6.1 by21211ˆσS n and22222ˆσS n leads to the following important conclusion.There is a special case when the two random samples are from a same normal population. Thus, 21σσ= and we have the result below.The proof is left as an exercise. F-propertiesFind )7,3(),4,5(95.001.0F F (15.52 , 0.1125)Using the result in the previous exercise, show that nm m n f f ,,1,,1αα-=Ifαα=>)},({m n F F P ，Thenαα-=≤1)},({m n F F P ，（1）),(~11n m F FF =， 11)},({1αα=>n m F F P 11}),(1{)},(1{)},({111αααα=<=>=>n m F F P n m F F P n m F F P （2） let,11αα-=),(1),(1m n F n m F αα=-。

概率论与数理统计(英文) 第二章

2. Probability （概率）2.1 Sample Space 样本空间statistical experiment (random experiment)----repeating----more than one outcome----know all the outcomes, but don ’t predict whichoutcome will be occurexample :toss an honest coin---- In this experiment there are only two possible outcomes:{head}, {tail}toss two honest coins---- In this experiment there are 4 possible outcomes:{H, H}, {H, T}, {T, H}, {T, T}Each outcome in a sample space is called a sample point of the sample space.Example 2.1.1 Consider the experiment of tossing a die. If we are interested in the number thatshows on the top face, the sample space would be{}11,2,3,4,5,6S =If we are interested only in whether the numbers is even or odd, the sample space is simply{}2,S even odd =Example 2.1.3 An experiment consists of flipping a coin and then flipping it a second time if ahead occurs. If a tail occurs on the first flip then a die is tossing once. To list the elements of thesample space providing the most information,we construct a diagram of Fig 2.1.1, which is called a tree diagram. Now the various paths alongthe branches of the tree give the distinct sample points. Starting with the top left branch andmoving to the right along the first path, we get the sample point HH, indicating the possibility thatheads occurs on two successive flips of the coin. The possibility that coin will show a tail followedby a 4 on the toss of the die is indicated by T4. Thus the sample space is{,,1,2,3,4,5,6}S HH HT T T T T T T =Fig .2.1.1 Tree diagram for Example 2.1.3 Definition 2.2.1 An event is a subset of a sample space.Example 2.2.1 Given the sample space {|0}S t t =≥. where t is the life in hours of a certainbulb, we are interest in the event B that a bulb burnt out before 200 hrs, i.e. the subset{|0200}B t t =≤< of S .Example 2.2.2 Assume that the unemployment rate r of a region is between 0 and 15%，i.e. wehave the sample space {|00.15}S r r =≤≤. If the event C “unemployment rate is low ”means that 0.04r ≤, then we have the subset {|00.04}C r r =≤≤ of S .You may have known operation of subsets, i.e.the complement of a subset （余集）,the union of subset,(并集)the difference of subset （差集）intersection of subsets （交集）,so we can say about the complement of an event, the union, difference and intersection ofevents.certain event （必然事件）：The sample space S itself, is certainly an event, which is called a certain event, means thatit always occurs in the experiment.impossible event （不可能事件）：The empty set, denoted by ∅, is also an event, called an impossible event, means that it neveroccurs in the experiment.Example 2.2.3 Consider the experiment of tossing a die, then{1,2,3,4,5,6}S =Let x be the number that shows on the top face, then the event {|,10}A x x S x =∈≤, isthe certain event, i.e. A S =.Then even {|,B x x S x =∈ is an irrational number }, (irrational-无理数)is the impossible event, i.e.B =∅.Let be the event consisting of all even numbers and be the event consisting of numbers divisible by 3. Find A , ,,A B A B A B .Solution We have {2,4,6,8},{3,6,9}.A B ==Thus{1,3,5,7,9}A = {2,3,4,6,8,9}A B = {6}A B = {3,9}A B = The relationship between events and the corresponding sample space can be illustrated graphical by means of Venn diagrams. In a Venn diagram, we represent the sample space by a rectangle and represent events by circles drawn inside the rectangle.Example 2.2.5 In Figure 2.2.1A B = regions 1 and 2, A D = regions 1,2 ,3 ,4 ,5 and 7,A B D = regions 2, 6 and 7A B D 1234567Fig 2.2.1 Venn diagram of Example 2.2.5The following list summarizes the rules of the operations of events.1. A ∅=∅2. A A ∅=3. A A A =4. A A =∅5. A A S =6. S =∅7. S ∅=8. A A =9. _________A B A B =10. _________A B A B =11. AB B A = 12. ()()AB C A B C = 13. ()()()AB C A C B C = 14. ()()()A B C A C B C =2.3 Probability of events1. relative frequency --------probability用频率定义概率Considering an Example.We plant 100 untreated cotton seeds.If 49 seeds germinate, that is, if there are 49 success (by success in statistics we mean the occurrence of the event under discussion) in 100 trials, we say that the relative frequency of success is 0.49.If we plant more and more seeds, a whole sequence of values for the respective relative frequencies is obtained. In general, these relative frequencies approach a limit value, we call this limit the probability of success in a single trial. From the data of Table 2.3.1 it appears that the relative frequencies are approaching the value 0.51, which we call the probability of a cotton seedling emerging from an untreated seed.relative frequencies are approaching the value 0.5252.0→n salways 1. Similarly, the probability of the impossible event is 0, and the probability of any event is always between 0 and 1.Note. In this definition , the word“limit”has a meaning which is different from the meaning you may have learned in calculus. We will discuss this problem later.Example 2.3.1Select 200 bulbs produced by company X at random of them 150 having life longer than 300hrs. Find the probability that the bulbs produced by company X have life longer than 300hrs.Solution.1500.75200p==Jixie-9-42.”equally likely to occur”------probability（古典概率，有限、等可能性）In many cases, the probability may be stated without experience. If we toss a properly balanced coin, we believe that the probability of getting a head is 0.5. We make this statement since in tossing a properly balanced coin, only two outcomes are possible and both outcome areIt should be pointed out that this definition is in a sense circular in nature, since the expression “equally likely to occur”itself involves the idea of probability. However, since this term is generally intuitively understood, the concept of “equally likely” will be left undefined.Solution.The sample space S consists of 20 sample points. The event {A= a white ball is drawn}consists of 4 sample points, thus the probability of drawing a white ball is4()0.2P A==.Solution The sample space is{S =(,)|,m n m n are positive integers 6}≤thus S consists of 36 sample points .Let {A =getting a total of 9},{B =getting a total greater of 9}Then we have{(3,6),(4,5),(5,4),(6,3)}A ={(4,6),(5,5),(6,4),(5,6),(6,5),(6,6)}B =Thus41()369P A == , 61()366P B ==. where ()p A is the probability that the event A occurs ./*******/Example 2.3.4 抽球问题Consecutively draw ballFrom a big pack which contains a white balls and b black balls, a ball is consecutively draw at random. what is the probability that the ball which be drawn in m-th time is white?抽球问题袋中有a 个白球，b 个黑球，从中依次任取一个球，且每次取出的球不再放回去，求第m 次取出的球是白球的概率。

(完整word)概率论与数理统计英文版总结,推荐文档

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.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 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 that the event A consists of n sample points, then the probability p that A occurs is()np P AN==Mutually exclusive(互斥事件)Two events A and B are said to be independent if()()()P A B P A P B=⋅IOr Two events A and B are independent if and only if(|)()P B A P B=.Conditional Probability 条件概率The probability of an event is frequently influenced by other events.If 12k ,,,A A A L are events, then12k 121312121()()(|)(|)(|)k k P A A A P A P A A P A A A P A A A A -=⋅⋅I I L I L I I L I Ifthe events12k ,,,A A A L areindependent, then for any subset12{,,,}{1,2,,}m i i i k ⊂L L ,1212()()()()m m P A A A P A P A P A i i i i i i =I I L L(全概率公式 total probability)()(|)()i i P B A P B A P A =IUsing the theorem of total probability, we have1()(|)(|)()(|)i i i kjjj P B P A B P B A P B P A B ==∑ 1,2,,i k =L1. random variable definition2. Distribution functionNote 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 following properties:3.2 Discrete Random Variables 离散型随机变量geometric distribution （几何分布）Binomial distribution（二项分布）poisson distribution（泊松分布）Expectation (mean) 数学期望2．Variance 方差standard deviation （标准差）probability density function概率密度函数5. Mean （均值）6. variance 方差.4.2 Uniform Distribution 均匀分布The uniform distribution, with the parameters a a nd b , has probability density function1for ,()0 elsewhere,a xb f x b a⎧<<⎪=-⎨⎪⎩4.5 Exponential Distribution 指数分布4.3 Normal Distribution正态分布1. Definition4.4 Normal Approximation to the Binomial Distribution （二项分布）4.7 C hebyshev’s Theorem （切比雪夫定理）Joint 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 space, the joint probabilitydistribution 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. 5.2 C onditional 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(,)(|)()X Y p x y p x y p y =for all x provided ()0Y p y ≠. 5.3 S tatistical independent 随机变量的独立性5.4 Covariance 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.理We can find the steadily of the frequency of the events in large number of random phenomenon. And the average of large number of random variables are also steadiness. These results are the law of large numbers.population (总体)A population may consist of finitely or infinitely many varieties. sample （样本、子样）中位数Sample Distributions 抽样分布1．sampling distribution of the mean 均值的抽样分布It is customary to write )(X E as X μ and )(X D as 2X σ.Here, ()E X μ= is called the expectation of the mean .均值的期望 n X σσ= is called the standard error of the mean. 均值的标准差7.1 Point Estimate 点估计Unbiased estimator(无偏估计量)minimum variance unbiased estimator （最小方差无偏估计量）3. Method of Moments 矩估计的方法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 functionmaximum likelihood estimate（最大似然估计）8.1 Statistical Hypotheses（统计假设）显著性水平Two Types of Errors。

《概率论与数理统计》(全英语)教学大纲课程名称概率

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概率论与数理统计英文版总结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.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 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 that the event A consists of n sample points, then the probability p that A occurs is()np P AN==Mutually exclusive(互斥事件)Mutually independent 事件的独立性Two events A and B are said to be independent if ()()()P A B P A P B=⋅IOr Two events A and B are independent if and only if(|)()P B A P B=.Conditional Probability 条件概率The multiplication theorem 乘法定理 If 12k ,,,A A A L are events, then12k 121312121()()(|)(|)(|)k k P A A A P A P A A P A A A P A A A A -=⋅⋅I I L I L I I L I If the events 12k ,,,A A A L are independent, then for any subset12{,,,}{1,2,,}m i i i k ⊂L L ,1212()()()()m m P A A A P A P A P A i i i i i i =I I L L(全概率公式 total probability)（贝叶斯公式Bayes ’ formula.）()(|)()i i P B A P B A P A =IUsing the theorem of total probability, we have1()(|)(|)()(|)i i i kjjj P B P A B P B A P B P A B ==∑ 1,2,,i k =L1. random variable definition2. Distribution functionNote 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 following properties:3.2 Discrete Random Variables 离散型随机变量geometric distribution （几何分布）Binomial distribution（二项分布）poisson distribution（泊松分布）Expectation (mean) 数学期望2．Variance 方差 standard deviation （标准差）probability density function概率密度函数5. Mean（均值）6. variance 方差4.2 Uniform Distribution 均匀分布The uniform distribution, with the parameters a a nd b , has probability density function1for ,()0 elsewhere,a x b f x b a⎧<<⎪=-⎨⎪⎩4.5 Exponential Distribution 指数分布4.3 Normal Distribution 正态分布4.4 Normal Approximation to the Binomial Distribution （二项分布）4.7 Chebyshev’s Theorem （切比雪夫定理）Joint 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 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. 5.2 C onditional 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(,)(|)()X Y p x y p x y p y =for all x provided ()0Y p y ≠.5.3S tatistical independent 随机变量的独立性5.4 Covariance 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.理We can find the steadily of the frequency of the events in large number of random phenomenon. And the average of large number of random variables are also steadiness. These results are the law of large numbers.population (总体)sample （样本、子样）中位数Sample Distributions 抽样分布1．sampling distribution of the mean 均值的抽样分布It is customary to write )(X E as X μ and )(X D as 2X σ.Here, ()E X μ= is called the expectation of the mean .均值的期望 n X σσ= is called the standard error of the mean. 均值的标准差7.1 Point Estimate 点估计Unbiased estimator(无偏估计量)minimum variance unbiased estimator（最小方差无偏估计量）3. Method of Moments 矩估计的方法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 functionmaximum likelihood estimate（最大似然估计）8.1 Statistical Hypotheses（统计假设）显著性水平Two Types of Errors。