Lecture 4 - Probability and Sampling Theory, Normal Distribution, and z

Probability and Statistics Course - English Version IntroductionProbability and statistics are important fields of studyin mathematics that deal with the analysis of data and the likelihood of certn events. This course is designed toprovide a solid foundation in both probability and statistics, with an emphasis on understanding the concepts and their applications in real-world situations.Course ObjectivesThe objectives of this course are as follows:1.To understand the basic principles of probabilityand statistics2.To learn how to use probability distributions tomake predictions and analyze data3.To develop an understanding of statisticalinference and hypothesis testing4.To become familiar with various statisticalsoftware tools and their applicationsCourse OutlineThe course is divided into several modules, each covering a different aspect of probability and statistics. The following is an overview of the modules:Module 1: Introduction to ProbabilityThis module introduces the fundamental principles of probability, including the definitions of probability, independence, conditional probability, Bayes’ rule, and random variables. Topics covered in this module include: •Basic Probability Concepts•Random Variables and Probability Distributions•Expected Value and Variance•Joint Probability Distributions•Conditional Probability and Bayes’ Rule•IndependenceModule 2: Discrete Probability DistributionsThis module focuses on the different types of discrete probability distributions, including the binomial distribution, the Poisson distribution, and the geometric distribution. Topics covered in this module include:•Discrete Probability Distributions•Binomial Distribution•Poisson Distribution•Geometric Distribution•Hypergeometric Distribution•Negative Binomial DistributionModule 3: Continuous Probability DistributionsThis module covers the different types of continuous probability distributions, including the normal distribution, the exponential distribution, and the Weibull distribution. Topics covered in this module include:•Continuous Probability Distributions•Normal Distribution•Standard Normal Distribution•Exponential Distribution•Weibull DistributionModule 4: Sampling and Statistical InferenceThis module covers statistical inference and hypothesis testing, including the central limit theorem, confidence intervals, and hypothesis testing for means and proportions. Topics covered in this module include:•Sampling and Sampling Distributions•Central Limit Theorem•Confidence Intervals•Hypothesis Testing for Means and Proportions•Type I and Type II Errors•Power and Sample SizeModule 5: Regression and Correlation AnalysisThis module covers the basics of regression andcorrelation analysis, including simple linear regression, multiple linear regression, and correlation analysis. Topics covered in this module include:•Simple Linear Regression•Multiple Linear Regression•Correlation Analysis•Nonlinear Regression•Model SelectionCourse RequirementsTo successfully complete this course, you will need access to the following:• A computer with internet access•Access to statistical software packages such as R, SPSS, or SAS (or similar software)• A basic understanding of algebra and calculus EvaluationEvaluation will be based on the following components: •Class participation and attendance•Homework assignments•Quizzes and examinationsConclusionBy the end of this course, you should have a thorough understanding of probability and statistics and their applications in real-world situations. Whether you plan to pursue further studies in mathematics, or use this knowledge in your career, this course will provide you with a solid foundation in this important field.。

学科英文知识点总结Introduction:In the expansive world of academia, there are countless disciplines and subjects that cover a wide range of topics and areas of study. Each subject has its own unique set of knowledge and principles that students are expected to learn and understand. This summary will discuss the key concepts in various academic subjects, including mathematics, science, history, literature, and more. By understanding these fundamental concepts, students can build a solid foundation for further learning and advancement in their academic pursuits.Mathematics:Mathematics is a fundamental subject that teaches students about the principles of numbers, shapes, and logical reasoning. Some key concepts in mathematics include:1. Arithmetic: Arithmetic is the most basic form of mathematics that deals with the basic operations of addition, subtraction, multiplication, and division.2. Algebra: Algebra introduces students to the concept of variables and how they can be used in equations to solve for unknown quantities.3. Geometry: Geometry focuses on the study of shapes, angles, and spatial relationships, including the properties of triangles, circles, and polygons.4. Calculus: Calculus is a branch of mathematics that deals with the study of rates of change and accumulation, including concepts such as derivatives and integrals.5. Probability and Statistics: Probability and statistics involve the study of chance and data analysis, including concepts such as probability distributions, sampling, and hypothesis testing.Science:Science encompasses the study of the natural world, including the principles of biology, chemistry, physics, and environmental science. Some key concepts in science include:1. Cell Theory: Cell theory is the principle that all living organisms are composed of cells, and that cells are the basic unit of life.2. Periodic Table: The periodic table is a systematic arrangement of the chemical elements, organized by their atomic number and electron configuration.3. Newton's Laws of Motion: Newton's laws of motion describe the relationship between the motion of an object and the forces acting on it, including concepts such as inertia, acceleration, and action-reaction.4. Evolution: Evolution is the process by which living organisms gradually change and adapt to their environment over time, as described by the theory of natural selection.5. Climate Change: Climate change refers to the long-term alteration of temperature and typical weather patterns in a place.History:History is the study of past events, cultures, and societies, and how they have influenced the present and future. Some key concepts in history include:1. The Industrial Revolution: The Industrial Revolution was a period of significant economic, technological, and social change that began in the late 18th century and transformed how goods were produced and distributed.2. World Wars: The two world wars were global conflicts that resulted in significant political, economic, and social changes, and had a profound impact on the course of world history.3. Civil Rights Movement: The civil rights movement was a social and political movement in the United States that sought to end racial segregation and discrimination against African Americans.4. Colonialism: Colonialism refers to the policy of a nation seeking to extend or retain its authority over other people or territories.5. The Renaissance: The Renaissance was a period of European history that marked the transition from the medieval period to the modern age, characterized by a renewed interest in art, literature, and science.Literature:Literature encompasses the study of written works, including novels, poetry, drama, and literary criticism. Some key concepts in literature include:1. Characterization: Characterization is the process of creating and developing a character, including their personality, motivations, and relationships with other characters.2. Themes: Themes are the central ideas or messages that are conveyed in a literary work, such as love, death, power, or freedom.3. Symbolism: Symbolism is the use of symbols to represent ideas or qualities, such as usinga rose to symbolize love or a dove to symbolize peace.4. Literary Devices: Literary devices are techniques used by writers to create a particular effect, such as metaphor, simile, irony, and foreshadowing.5. Literary Movements: Literary movements are periods or styles of literature that share common characteristics, such as romanticism, realism, modernism, and postmodernism. Conclusion:Academic subjects cover a vast array of knowledge and principles that are essential for students to understand in order to succeed in their studies. By grasping the key concepts in mathematics, science, history, literature, and other subjects, students can build a solid foundation for further learning and critical thinking, and are better prepared to navigate the complex world of academia.。

Probability Theory and Examples 第四版课程设计选题背景概率论是数学中的一个重要分支，旨在研究随机现象的规律性。

作为一门学科，概率论在现代科学中有着广泛的应用，从统计学到生物学、物理学、经济学都需要用到概率论知识。

因此，学习概率论是非常有意义的。

本课程设计旨在通过掌握“Probability Theory and Examples”第四版的相关知识，提高学生对概率论的理解，并为未来的学习提供基础。

教学目标本课程设计旨在达到以下教学目标：1.熟悉概率论的基本概念和方法。

2.了解随机变量和概率分布。

3.掌握概率分布的各种性质和应用。

4.熟悉极限定理和大数定理的原理和应用。

5.熟悉随机过程的基本概念和分类。

6.掌握随机过程的应用问题。

教学内容第一章概率论1.1 随机试验和概率论基本概念1.2 频率学派和贝叶斯学派第二章随机变量和概率分布2.1 随机变量和概率分布2.2 散点图、直方图和箱线图第三章多维随机变量和相关性3.1 多维随机变量和相关性3.2 协方差和相关系数第四章极限定理和大数定理4.1 正态分布和中心极限定理4.2 大数定理和中心极限定理的应用第五章随机过程5.1 随机过程和马尔可夫性5.2 布朗运动和随机游走第六章马尔可夫链的预备知识6.1 马尔可夫链和平稳分布6.2 马尔可夫链的转移概率矩阵第七章应用问题7.1 应用问题的分析和解决7.2 系统可靠性评估教学方法本课程使用课堂讲授、案例分析和实践练习相结合的教学方法，根据不同章节的内容特点，采用相应的教学方法，如分组讨论、探究式学习、问题导向。

教学评价本课程的教学评价分为两个方面：学生对知识点的掌握情况和学生的应用能力。

教师将通过课堂测试、期中期末考试、课堂作业等形式对学生的知识点掌握情况进行评估。

同时，引入案例解析和实践练习，提高学生的应用能力。

教材推荐“Probability Theory and Examples”第四版，作者Richard Durrett。

《Business Statistic》中国人民大学出版社英文版第五版chapter1~8复习参考Part1名词解释1、Statistics is a method of extracting useful information from a set of numerical data in orderto make a more effective and informed decision.2、Descriptive Statistics：These are statistical methods of organizing, summarizing andpresenting numerical data in convenient forms such as graphs, charts and tables.3、Inferential statistics is defined as statistical methods used for drawing conclusions about apopulation based on samples.4、Primary data is obtained first hand.5、Secondary data already exists or has been previously collected such as company accounts, orsales figures.6、Mean: The arithmetic average and the most common measure ofaaaaaaa central tendency.①All values are included in computing the mean.②A set of data has a unique mean ③Themean is affected by unusually large or small data points (outliers / extreme values).7、Mode: The most frequent data, or data corresponding to the highest frequency. ①Mode isnot affected by extreme values. ②There may not be a mode. ③There may be several modes. ④Used for either numerical or categorical data.8、Median is the value that splits a ranked set of data into two equal parts. ①Median is notaffected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur.9、Standard Deviation: ①A measure of the variation of data from the mean. ②The mostcommonly used measure of variation. ③Represented by the symbol ‘s’. ④Shows how the data is distributed around the mean.10、Probability is the chance of an occurrence of an event. ①Probability of an eventalways lies between 0 and 1. ②The sum of the probabilities of every possible outcome or event is 1. ③The probability of the complement A’ is given by 1-P(A).11、Properties of Normal distribution：①Continuous random variable. ②‘Bell-shaped’ &symmetrical. ③Mean, median, mode are equal ④Area under the curve is 1.12、The Central Limited Theorem：①If the population followed normal distribution, thesampling distribution of mean is followed normal distribution. ②If the population do not followed normal distribution, but the sample size is larger than 30, the sampling distribution of mean is followed normal distribution.Part2选择题Topic 1 - Introduction to Business Statistics & Data CollectionQ1. The universe or totality of items or things under consideration is called:a. a sample.b. a population.c. a parameter.d.none of the above.Q2. Those methods involving the collection, presentation, and characterization of a set of data in order to properly describe the various features of that set of data are called:a.inferential statistics.b.total quality management.c.sampling.d.descriptive statistics.Q3. The portion of the universe that has been selected for analysis is called:a. a sample.b. a frame.c. a parameter.d. a statistic.Q4. A summary measure that is computed to describe a numerical characteristic from only a sample of the population is called:a. a parameter.b. a census.c. a statistic.d.the scientific method.Q5. A summary measure that is computed to describe a characteristic of an entire population is called:a. a parameter.b. a census.c. a statistic.d.total quality management.Q6. The process of using sample statistics to draw conclusions about population parameters is called:a.inferential statistics.b.experimentation.c.primary sources.d.descriptive statistics.Q7. Which of the four methods of data collection is involved when a person retrieves data from an online databasea.published sources.b.experimentation.c.surveying.d.observation.Q8. Which of the four methods of data collection is involved when people are asked to complete a questionnairea.published sources.b.experimentation.c.surveying.d.observation.Q9. Which of the four methods of data collection is involved when a person records the use of the Los Angeles freeway systema.published sources.b.experimentation.c.surveying.d.observation.Q10. A focus group is an example of which of the four methods of data collectiona.published sources.b.experimentation.c.surveying.d.observation.Q11. Which of the following is true about response ratesa.The longer the questionnaire, the lower the rate.b.Mail surveys usually produce lower response rates than personal interviews or telephonesurveys.c.Question wording can affect a response rate.d. d. All of the above.Q12. Which of the following is a reason that a manager needs to know about statisticsa.To know how to properly present and describe information.b.To know how to draw conclusions about the population based on sample information.c.To know how to improve processes.d.All of the above.Scenario 1-1Questions 13-15 refer to this scenario:An insurance company evaluates many variables about a person before deciding on an appropriate rate for automobile insurance. Some of these variables can be classified as categorical, discrete and numerical, or continuous and numerical.Q13. Referring to Scenario 1-1 (above), the number of claims a person has made in the last three years is what type of variablea.Categorical.b.Discrete and numerical.c.Continuous and numerical.d.None of the above.Q14. Referring to Scenario 1-1 (above), a person's age is what type of variablea.Categorical.b.Discrete and numerical.c.Continuous and numerical.d.None of the above.Q15. Referring to Scenario 1-1 (above), a person's gender is what type of variablea.Categorical.b.Discrete and numerical.c.Continuous and numerical.d.None of the above.Q16. Which of the following can be reduced by proper interviewer traininga.Sampling error.b.Measurement error.c.Coverage error.d.Nonresponse error.Scenario 1-2Questions 17-19 refer to this scenario:Mediterranean fruit flies were discovered in California a few years ago and badly damaged the oranges grown in that state. Suppose the manager of a large farm wanted to study the impact of the fruit flies on the orange crops on a daily basis over a 6-week period. On each day a random sample of orange trees was selected from within a random sample of acres. The daily average number of damaged oranges per tree and the proportion of trees having damaged oranges were calculated.Q17. Referring to Scenario 1-2 (above), the two main measures calculated each day ., average number of damaged oranges per tree and proportion of trees having damaged oranges) are called _______.a.statistics.b.parameters.c.samples.d.populations.Q18. Referring to Scenario 1-2 (above), the two main measures calculated each day ., average number of damaged oranges per tree and proportion of trees having damaged oranges) may be used on a daily basis to estimate the respective true population _______.a.estimates.b.parameters.c.statistics.d.frame.Q19. Referring to Scenario 1-2 (above), in this study, drawing conclusions on any one day about the true population characteristics based on information obtained from the sample is called _______.a.evaluation.b.descriptive statistics.c.inferential statistics.d.survey.Scenario 1-3Questions 20 and 21 refer to this scenario:The Quality Assurance Department of a large urban hospital is attempting to monitor and evaluate patient satisfaction with hospital services. Prior to discharge, a random sample of patients is asked to fill out a questionnaire to rate such services as medical care, nursing, therapy, laboratory, food, and cleaning. The Quality Assurance Department prepares weekly reports that are presented at the Board of Directors meetings and extraordinary/atypical ratings are easy to flag.Q20. Referring to Scenario 1-3 (above), true population characteristics estimated from thesample results each week are called _____________.a.inferences.b.parameters.c.estimates.d.data.Q21. Referring to Scenario 1-3 (above), a listing of all hospitalised patients in this institution over a particular week would constitute the ________.a.sample.b.population.c.statistics.d.parameters.Scenario 1-4Questions 22-24 refer to this scenario:The following are the questions given to Sheila Drucker-Ferris in her college alumni association survey. Each variable can be classified as categorical or numerical, discrete or continuous.Q22. Referring to Scenario 1-4 (above), the data for the number of years since graduation is categorised as: __________________.a.numerical discrete.b.categorical.c.numerical continuous.d.none of the above.Q23. Referring to Scenario 1-4 (above), the data for the number of science majors is categorised as: ____________.a.categorical.b.numerical continuous.c.numerical discrete.d.none of the above.Q24. Referring to Scenario 1-4 (above), the data for tabulating the level of job satisfaction (High, Moderate, Low) is categorised as: _________.a.numerical continuous.b.categorical.c.numerical discrete.d.none of the above.Topic 2: Organising and Presenting dataQ1 The width of each bar in a histogram corresponds to the:a.boundaries of the classes.b.number of observations in the classes.c.midpoint of the classes.d.percentage of observations in the classes.Q2 When constructing charts, which of the following chart types is plotted at the class midpointsa.Frequency histograms.b.Percentage polygons.c.Cumulative relative frequency ogives.d.Relative frequency histograms.Q3 When polygons or histograms are constructed, which axis must show the true zero or "origin"a.The horizontal axis.b.The vertical axis.c.Both the horizontal and vertical axes.d.Neither the horizontal nor the vertical axis.Q4 To determine the appropriate width of each class interval in a grouped frequency distribution, we:a.divide the range of the data by the number of desired class intervals.b.divide the number of desired class intervals by the range of the datac.take the square root of the number of observations.d.take the square of the number of observations.Q5 When grouping data into classes it is recommended that we have:a.less than 5 classes.b.between 5 and 15 classes.c.more than 15 classes.d.between 10 and 30 classes.Q6 Which of the following charts would give you information regarding the number of observations "up to and including" a given groupa.Frequency histograms.b.Polygons.c.Percentage polygons.d.Cumulative relative frequency ogives.Q7 Another name for an "ogive" is a:a.frequency histogram.b.polygon.c.percentage polygon.d.cumulative percentage polygon.Q8 In analyzing categorical data, the following graphical device is NOT appropriate:a.bar chart.b.Pareto diagram.c.stem and leaf display.d.pie chart.Table 2The opinions of a sample of 200 people broken down by gender about the latest congressionalQ9 Table 2 (above) contains the opinions of a sample of 200 people broken down by gender about the latest congressional plan to eliminate anti-trust exemptions for professional baseball. Referring to Table 2, the number of people who are neutral to the plan is _______.a.36b.54c.90d.200Q10 Referring to Table 2, the number of males who are against the plan is _______.a.12b.48c.60d.96Q11 Referring to Table 2, the percentage of males among those who are for the plan is ______.a.%b.24%c.25%d.76%Q12 Referring to Table 2, the percentage who are against the plan among the females is _______.a.%b.20%c.30%d.52%Topic 3: Numerical Descriptive StatisticsQ1 Which measure of central tendency can be used for both numerical and categorical variablesa.Mean.b.Median.c.Mode.d.Quartiles.Q2 Which of the following statistics is not a measure of central tendencya.Mean.b.Median.c.Mode.d.Q3.Q3 Which of the following statements about the median is NOT truea.It is more affected by extreme values than the mean.b.It is a measure of central tendency.c.It is equal to Q2.d.It is equal to the mode in bell-shaped distributions.Q4 The value in a data set that appears most frequently is called:a.the median.b.the mode.c.the mean.d.the variance.Q5 In a perfectly symmetrical distribution:a.the mean equals the median.b.the median equals the mode.c.the mean equals the mode.d.All of the above.Q6 When extreme values are present in a set of data, which of the following descriptive summary measures are most appropriatea.CV and range.b.Mean and standard deviation.c.Median and interquartile range.d.Mode and variance.Q7 The smaller the spread of scores around the mean:a.the smaller the interquartile range.b.the smaller the standard deviation.c.the smaller the coefficient of variation.d.All the above.Q8 In a right-skewed distribution:a.the median equals the mean.b.the mean is less than the median.c.the mean is greater than the median.d.the mean is less than the mode.a.b.c.d.Q10 Referring to Table 3 (above), the median carbohydrate amount in the cereal is ________ grams.a.19b.20c.21d.Q11 Referring to Table 3 (above), the 1st quartile of the carbohydrate amounts is ________ grams.a.15b.20c.21d.25Q12 Referring to Table 3 (above), the range in the carbohydrate amounts is ________ grams.a.16b.18c.20d.21Topic 4: Basics probability and discrete probability distributionsInformation A, needed to answer Questions 1 to 2The Health and Safety committee in a large retail firm is examining the relationship between the number of days of sick leave an employee takes and whether an employee works on the day shift (D) or night shift (N). The committee looks at a sample of 50 employees and notes which shift they work on and whether the number of days of sick leave they take in a year is less than 6 daysQ1 Use Information A to answer this question. Which of the following statements about the values in the table of probabilities is not correcta.The probability of an employee taking 6 or more days of sick leave P(M) isb.The probability that an employee is on the Night Shift (N) and takes less than 6 days ofleave (L), is called a conditional probability P(N | L) =c.If you know that an employee is on day shift (D) then the probability that they will takeless than 6 days of leave (L) is the conditional probability P(L | D) =d.The probability that an employee works Day Shift (D) or takes 6 or more days of leave (M)is found using the addition rule to be P(D or M) =e.They are all correctQ2 The analyst wishes to use the Probabilities table from Information A to determine whether the work shift variable and the number of days of sick leave variable are or are not independent variables. Which of the following statements about the work shift and the number of days of sick leave variables is correcta.These variables are independent because the marginal probabilities such as P(L) are thesame as the conditional probabilities P(L | D)b.These variables are not independent because the marginal probability P(L) is differentfrom the conditional probability P(N | L)c.These variables are not independent because the joint probabilities such as P(L and N)are equal to the product of the probabilities P(L).P(N).d.These variables are dependent because the marginal probabilities such as P(L) are equalto the conditional probability P(L | N)e.None of the aboveInformation B, needed to answer Question 3Suppose the manager of a home ware retailer decides in a 5-minute period no more than 4 customers can arrive at a counter. Using past records he obtains the following probabilitythe following is the correct pair of values for the mean, the variance or standard deviation of the number of arrivals at the counter.a.Mean mu = 2 and variance sigma-squared =b.Mean mu = and variance sigma-squared =c.Mean mu = 2 and standard deviation sigma =d.Mean mu = and variance sigma-squared =e.None of the aboveInformation C, needed to answer Questions 4-6The section manager in an insurance company is interested in evaluating how well staff at the inquiry counter handle customer complaints. She interviews a sample of n = 6 customers who have made complaints and asks each of them whether staff had handled their complaints well. Each interview is called a trial. If a customer says their complaint was handled well this is called a success. She thinks that as long as these people are interviewed independently of each other then the number of people who say their complaint was handled well is a random variable with a Binomial probability distribution. The section manager thinks that the probability that a customers complaint will be handled well is p = .Q4 Use Information C to answer this question. A total of n = 6 people are interviewed independently of each other. Which of the following statements about the probability that 5 out of the 6 complaints will be handled well is correcta.less thanb.between andc.more thand.between ande.None of the aboveQ5 Using Information C, which of the following statements about the probability that 4 or less of the 6 complaints will be handled well is correcta.less thanb.more thanc.between andd.between ande.None of the aboveQ6 Suppose the section manager from Information C is interested in the measures of central tendency and variation for the number of complaints which are handled well. Which of the following sets of values, where values are rounded to 3 decimal places, is the correct set of valuesa.Mean mu = and variance sigma-squared =b.Mean mu = and variance sigma-squared =c.Mean mu = and variance sigma-squared =d.Mean mu = and standard deviation sigma =e.None of the aboveInformation D, needed to answer Questions 7-9The manager of a large retailer thinks that one reason why staff at the complaints counter fail to handle customer complaints well is that not enough staff are allocated to this counter. Past experience has shown that the number of customers who arrive at this counter has a Poisson distribution where the average number who arrive each hour is 36. He decides to look at how many customers are likely to arrive at the complaints counter during a 5-minute period.Q7 Use Information D to answer this question. Which of the following statements concerning the probability that exactly 2 customers will arrive at the counter in a 5-minute period is correcta.less thanb.between andc.between andd.more thane.None of the aboveQ8 Use Information D to answer this question. Which of the following statements concerning the probability that 3 or more customers will arrive at a counter in a 5-minute period is correcta.between andb.less thanc.more thand.between ande.None of the aboveQ9 The section manager from Information D is interested in the mean and variance of the number of customers who arrive during a 1 hour period. Which of the following is the correct set of values for these two measuresa.Mean mu = 3 and variance sigma-squared = 3b.Mean mu = 36 and standard deviation sigma =c.Mean mu = 30 and variance sigma-squared = 30d.Mean mu = 36 and standard deviation sigma = 6e.None of the aboveTopic 5: Normal probability distribution & sampling distributionQ1 Which of the following is not a property of the normal distributiona.It is bell-shaped.b.It is slightly skewed left.c.Its measures of central tendency are all identical.d.Its range is from negative infinity to positive infinity.Q2 The area under the standardized normal curve from 0 to would be:a.the same as the area from 0 to .b.equal to .c.found by using Table in your textbook.d.all of the above.Q3 Which of the following about the normal distribution is not truea.Theoretically, the mean, median, and mode are the same.b.About two-thirds of the observations fall within ± 1 standard deviation from the mean.c.It is a discrete probability distribution.d.Its parameters are the mean and standard deviation.Q4 In its standardized form, the normal distribution:a.has a mean of 0 and a standard deviation of 1.b.has a mean of 1 and a variance of 0.c.has a total area equal to .d.cannot be used to approximate discrete binomial probability distributions.Q5 In the standardized normal distribution, the probability that Z > 0 is _______.a.b.c.d.cannot be found without more informationQ6 The probability of obtaining a value greater than 110 in a normal distribution in which the mean is 100 and the standard deviation is 10 is ______________ the probability of obtaining a value greater than 650 in a normal distribution with a mean of 500 and a standard deviation of 100.a.less thanb.equal to.c.greater thand.It is unknown without more information.Q7 The probability of getting a Z score greater than is ________.a.close tob.c. a negative numberd.almost zeroQ8 For some positive value of Z, the probability that a standardized normal variable is between 0 and Z is . The value of Z isa.b.c.d.Q9 For some value of Z, the probability that a standardized normal variable is below Z is . The value of Z isa.b.c.d.Q10 Given that X is a normally distributed random variable with a mean of 50 and a standard deviation of 2, the probability that X is between 47 and 54 isa.b.c.d.Q11 For some positive value of X, the probability that a standardized normal variable is between 0 and + is . The value of X isa.b.c.d.Q12 The owner of a fish market determined that the average weight for a catfish is pounds with a standard deviation of pounds. A citation catfish should be one of the top 2 percent in weight. Assuming the weights of catfish are normally distributed, at what weight (in pounds) should the citation designation be establisheda.poundsb.poundsc.poundsd.poundsQ13 Which of the following is NOT a property of the arithmetic meana.It is unbiased.b.It is always equal to the population mean.c.Its average is equal to the population mean.d.Its variance becomes smaller when the sample size gets bigger.Q14 The sampling distribution of the mean is a distribution of:a.individual population values.b.individual sample values.c.statistics.d.parameters.Q15 The standard deviation of the sampling distribution of the mean is called the:a.standard error of the sample.b.standard error of the estimate.c.standard error of the mean.d.All of the aboveQ16 According to the central limit theorem, the sampling distribution of the mean can be approximated by the normal distribution:a.as the number of samples gets "large enough."b.as the sample size (number of observations) gets "large enough."c.as the size of the population standard deviation increases.d.as the size of the sample standard deviation decreases.Q17 For a sample size of n=10, the sampling distribution of the mean will be normally distributed:a.regardless of the population's distribution.b.if the shape of the population is symmetrical.c.if the variance of the mean is known.d.if the population is normally distributedTopic 6: EstimationQ1 The interval estimate using the t critical value is ________ than the interval estimate using the z critical value.a.Narrowerb.The same asc.Widerd.More powerfulQ2 To estimate the mean of a normal population with unknown standard deviation using a small sample, we use the ______ distribution.a.'t'b.'Z'c.samplingd.alphaQ3 If the population does not follow a normal distribution, then to use the t distribution to give a confidence interval estimate for the population mean, the sample size should be:a.at least 5b.at least 30c.at least 100d.less than 30Q4 The 'z' value or 't' value used in the confidence interval formula is called the:a.sigma valueb.critical valuec.alpha valued.none of the aboveQ5 The 'z' value that is used to construct a 90 percent confident interval is:a.b.c.d.Q6 The 'z' value that is used to construct a 95 percent confidence interval is:a.b.c.d.Q7 The sample size needed to construct a 90 percent confidence interval estimate for the population mean with sampling error ± when sigma is known to be 10 units is:a.9b.32c.75d.107Q8 The t critical value approaches the z critical value when:a.the sample size decreasesb.the sample size approaches infinityc.the confidence level increasesd.the sample is smallQ9 The t-critical value used when constructing a 99 percent confidence interval estimate with a sample of size 18 is:a.b.c.d.Q10 The t-value that would be used to construct a 90 percent confidence interval for the mean with a sample of size n 36 would be:a.b.c.d.Q11 The value of alpha (two tailed) for a 96 percent confidence interval would be: a.b.c.d.Q12 When using the t distribution for confidence interval estimates for the mean, the degrees of freedom value is:a.nb.n-1c.n-2d.n %2B 1Q13 You would interpret a 90 percent confidence interval for the population mean as:a.you can be 90 percent confident that you have selected a sample whose interval doesinclude the population meanb.if all possible samples are selected and CI's are calculated, 90 percent of those intervalswould include the true population meanc.90 percent of the population is in that intervald.both A and B are trueQ14 From a sample of 100 items, 30 were defective. A 95 percent confidence interval for the proportion of defectives in the population is:a.(.2, .4)b.(.21, .39)c.(.225, .375)d.(.236, .364)Q15 A confidence interval was used to estimate the proportion of statistics students that are male. A random sample of 70 statistics students generated the following 90 percent confidence interval: , . Using the information above, what size sample would be necessary if we wanted to estimate the true proportion to within ± using 95 percent confidencea.240b.450c.550d.150整理人：阿桤。

Lecture18:Poisson Processes–Part IISTAT205Lecturer:Jim Pitman Scribe:Matias Damian Cattaneo<cattaneo@>18.1Compound Poisson DistributionWe begin by recalling some things from last lecture.Let X1,X2,...be independent and identically distributed random variables with dis-tribution F on R;that is:F(B)=P[X∈B]Let Nλbe a Poisson random variable with meanλ;that is:P[Nλ=k]=λn e−λn .Interpretation of L:Recall that Poisson point process←→counting measure,andwe haveN(B)=Nλi=11{X i∈B}. 18-1That is,N (B )is the number of values 1≤i ≤N λwith X i ∈B .ObserveN (R )=N λ∼P oisson (λ)What is the distribution of N (B )?Apply the previous theorem with X i replaced by 1{X i ∈B }.So we have E e itN (B ) =exp e it −1 L (B ) ,so N (B )∼P oisson (L (B )).More generally for B 1,B 2,...,B m ;m disjoint sets we can compute,by the same argu-ment,E e i P m k =1t k N (B k ) =m k =1E e it k N (B k ) ,and observe that the LHS is the multivariate characteristic function of the vector (N (B 1),N (B 2),...,N (B m ))at (t 1,t 2,...,t m ),and the RHS is the multivariate char-acteristic function of a collection of independent random variables with a Poisson distribution.Consequently,by the uniqueness theorem for multivariate characteris-tic function (see text)we conclude that N (B 1),N (B 2),...,N (B m )are independent Poisson variables.18.2Summary so farNow we summarize our work so far.Let X 1,X 2,...be i.i.d.F .Let N λ∼P oisson (λ),independent of X 1,X 2,....Let N (B )= Nλi =11{X i ∈B },the point process counting values in B up to N λ.Then(N (B ),B ∈Borel )is a Poisson random measure with mean measure L ,meaning that if B 1,...,B m are disjoint Borel sets,(N (B i ),1≤i ≤m )are independent with distributions P oisson (L (B i ))for 1≤i ≤m ,respectively.Example 18.2(From previous lecture)Let 0<T 1<T 2<...be a sum of indepen-dent Exponential (λ)variables.So N t = ∞i =11{T i ≤t }∼P oisson (λt ).Then we see that (N t ,0≤t ≤T )has the same distribution as (N [0,t ],0≤t ≤T )whereN [0,t ]=N λ i =11{X i ≤t }for X 1,X 2,...∼U [0,T ].This is an example of a famous connection between sums of exponentials and uniform order statistics.Examples can be found in many texts,including [1].These are Poisson tricks!18.3Computations with CPNow we discuss some computations with CP(L).Think about this:we have a Poisson scatter with mean intensity L,say X1,X2,...,X n.Letλ=L(R).We haveS=Nλi=1X i= xN(dx)and recall thatN(B)=Nλi=11{X i∈B}∼P oisson(L(B))and alsoN(·)=Nλ i=1δX i(·)f(x)N(dx)=Nλi=1f(X i)Now we compute(You check details):E[S]=E xN(dx) = x E[N(dx)]= xL(dx)V[S]=V xN(dx) =V ... =...= x2L(dx) Example18.3ConsiderL= iλiδX iN(·)= i N iδX i(·)where N i∼P oisson(λi)and as i varies these are independent.Now we have:S= xN(dx)= i x i N(x i)E[S]= i x iλi= xL(dx)V[S]= i x2iλi= x2L(dx)Theorem18.4(L-K)Every∞-divisible distribution on R is a weak limit of shifted CP distributions.Look at the characteristic function of a centered CP distribution to see something new:take S∼CP(L)and look at(S−E[S]).Assuming that |x|L(dx)<∞,we haveE e it(S−E[S]) =exp{−it E[S]}exp e itx−1 L(dx)=exp e itx−1−itx L(dx)andE (S−E[S])2 = x2L(dx)from before.Observe that this formula defines a characteristic function for every positive measure L on R with L(−1,1)c=0and 1−1x2L(dx)<∞.You can easily check this;see texts such as[1].This leads to the general L-K Formula.18.4More details on L´e vy MeasureDefinition18.5A measure L on R is a L´e vy measure if it has the following prop-erties:1.L{(−ε,ε)c}<+∞,for allε>0.2.L{0}=0.3. 1−1x2L(dx)<+∞.For such an L,σ2≥0,c∈R,define the L´e vy-Khinchine exponent in the following way:ΨL,σ2,c(t)= e itx−1−itτ(x) L(dx)−12.eΨ(t)determines L,σ2,c uniquely.Before we prove this theorem,we consider a few examples.Example18.7 1.Consider a point massδc at c.Its characteristic function ise itc,and we see that itc=Ψ0,0,c(t).2.Consider now a normal distribution N(c,σ2).Its characteristic function ise itc−σ2t2/2and it is easy to see thatΨ(t)=itc−σ2t2/2corresponds to(0,σ2,c).3.Now,let N be a Poisson random measure.For each f≥0,we haveE e−θR fdN =exp e−θf(x)−1 µ(dx)Ifµis bounded measure,takeθ=−it,E e it R fdN =exp e itf(x)−1 µ(dx) .Let L(dy)=µ{x:f(x)∈dy}(restricted to{0}c).For those who doesn’tlike to see dy’s outside the integral sign,the definition of L could be L(B):=µ(f−1(B)).Then E e it R fdN =exp (e ity−1)L(dy) .Here we can recognize the enemy from the beginning of the lecture,and the characteristic function of fdN is exp(ΨL,0,c)where c= τ(x)L(dx).Proof:First,we will prove that eΨ(t)is a characteristic function,and the infinite divisibility is obvious(n-th root isΨ(L/n,σ2/n,c/n)).Fix t.Observe that for|x|<1we havee itx−1−itτ(x)=e itx−1−itx≤cx2t2(18.1) for|xt|small.Therefore,the integral converges because 1−1x2L(dx)<+∞and L{(−ε,ε)c}<+∞.HenceΨ(t)is a well-defined complex number for all t∈R. Second,since the product of characteristic functions is also a characteristic functionwe may assume without loss of generality thatσ2and c are both0.Let L n be L restricted to −1n c.Note that exp(ΨL n,0,0(t))is a characteristic function:since L n isfinite,exp(ΨL n,0,0(t))is the characteristic function of a shifted compound Poisson variable with parameter L n.From18.1and the dominated convergence theorem we see thatΨL n,0,0(t)=ΨL,0,0(t).limn→∞Since exp is continuous function we immediately have that exp(ΨL n,0,0(t))→exp(ΨL,0,0(t)) and it only remains to prove thatΨ(t)is continuous at0(in order to apply the L´e vy continuity theorem).This is left as an exercise for the reader.(The same dominated convergence theorem will work.)References[1]Richard Durrett.Probability:theory and examples,3rd edition.ThomsonBrooks/Cole,2005.。

ap统计学知识点总结In this article, we will provide a comprehensive overview of the key topics and concepts covered in AP Statistics, including:1. Exploring data: This topic covers the basics of collecting, organizing, and summarizing data. Students learn about different types of data, such as categorical and numerical data, and how to create visual displays, such as histograms and box plots, to effectively communicate the information contained in the data.2. Sampling and experimentation: This section introduces students to the principles of sampling and experimental design. Students learn about various sampling methods, such as simple random sampling and stratified sampling, and how to critically evaluate the design of statistical studies.3. Probability and random variables: This topic covers the fundamental principles of probability, including the concepts of independence, conditional probability, and expected value. Students also learn about different probability distributions, such as the binomial and normal distributions, and how to use them to analyze real-world data.4. Inference: This section covers the process of making inferences about population parameters based on sample data. Students learn about confidence intervals and hypothesis testing, and how to use them to draw conclusions about a population.5. Regression analysis: This topic introduces students to the principles of linear regression and correlation analysis. Students learn how to use regression models to analyze the relationship between two variables and make predictions based on the model.In addition to these main topics, the AP Statistics course also covers a range of other important concepts, including statistical inference, experimental design, and the interpretation of results. Throughout the course, students are expected to use statistical software and technology to analyze and interpret data, and to communicate their findings effectively through written reports and presentations.By the end of the course, students are expected to have developed a deep understanding of statistical concepts and techniques, and to be able to apply their knowledge to real-world situations. The course also aims to prepare students for the AP Statistics exam, which includes a multiple-choice section and a free-response section that assesses students' ability to analyze and interpret data, as well as their understanding of statistical concepts and methods.Overall, AP Statistics is a challenging and rewarding course that provides students with the knowledge and skills required to understand and interpret data. By studying the key topics and concepts covered in the course, students can develop the analytical and critical thinking skills required to succeed in a wide range of academic and professional fields.。