概率统计英文讲义
概率与统计(英文)chapter 1(2011) Probability and Statistic
The probability & statistics is a science of studying statistic law of random phenomena . This science generated from 17th century, it comes of gambling and is applied in gambling.But now it is the foundation of many sciences, for example, econometrics, control science, information science, decision theory, game theory. Especially, in finance and accounting.
The set of measurements collected for a particular element is called an observation.
• Branched of Statistics
Descriptive statistics
An investigator who has collected data may wish simply to summarize and describe important features of the data. This entails using methods from descriptive statistics. Some of these methods are graphical in nature; the construction of histograms, boxplots, and scatter plots are primary examples. Other descriptive methods involve calculation of numerical summary measures, such as means, standard deviations, and correlations coefficients
概率统计(英文) chapter5 Joint Probability Distributions and Random Samples
5.1 Jointly Distributed Random Variables 5.2 Expected Values, Covariance, and Correlation 5.3 Statistics and Their Distributions 5.4 The Distribution of the Sample Mean 5.5 The Distribution of a Linear Combination
200
100
0.20
0.10
0.20
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x
250
0.05
0.15
0.30
Determine P(Y≥100)=
Definition: The marginal probability mass functions of X and Y,
denoted by pX(x) and PY(y), respectively, are given by
f(x,y)
y
Surface f(x,y)
A=Shaded rectangle x
Example 5.3 A bank operates both a drive-up facility and a walk-up window. On a randomly selected day,
let X=the proportion of time that the drive-up facility is in use
p[(X,Y)A]f(x,y)dxdy
A
In particular, if A is the two-dimensional rectangle {(x,y):a≤x≤b, c≤y≤d},then
概率统计课件【英文】
Event – set of outcomes
Venn Diagrams
Outcomes are mutually exclusive – disjoint
S
1
4 Event A
2 5
3
6
Outcomes
An Example from Card Games
What is the probability of drawing two of the same card in a row in a shuffled deck of cards?
Communications Speech and Image Processing Machine Learning Decision Making Network Systems Artificial Intelligence
Used in many undergraduate courses (every grad course)
By their average behavior By the likelihood of particular outcomes
Allows us to build models for many physical behaviors
Speech, images, traffic …
Applications
Introduction to Probability: Counting Methods
Rutgers University Discrete Mathematics for ECE 14:332:202
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(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。
概率统计英文讲义Walpole02 PPT资料共23页
•A = ? •A = ? • A A’ = ? • A A’ = ? • S’ = ? • ’ = ? • (A’)’ = ? • (A B)’ = ? • (A B)’ = ? • (A B)’=A’ B’, (A B)’=A’ B’
• Why is this important? • This fact allows us to check our answers.
Properly enumerating the sample space is key to correctly calculating probabilities.
Two events are mutually exclusive if…?
•A B =
The union of two events, A and B?
• The set of elements in either A or B. A B = ? • A B = {HHH, HHT, HTH, HTT , THT, TTH}.
Walpole Chபைடு நூலகம்02: 9
Circular permutations: n distinct objects arranged in a circle.
• The position of the first object could be anywhere.
• If all objects moved one position clockwise, it’s still the same permutation.
The intersection of two events A and B?
概率统计英文讲义Walpole02
Suppose that we will give 3 different awards to three students out of a class of 60 students. How many ways can the awards be given.
• What if the problem was slightly different and one student could win all three awards?
Steve Kennedy
Walpole Ch 02: 1
Probability Intro
Walpole Ch 02: 2
The sample space S of an experiment is the set of all possible outcomes.
• We must understand the sample space in order to determine the probability of each outcome occurring.
• Another way to look at it: the set of all permutations is n!. For each starting value, there are n orderings that are identical (moving the same ordering around the circle). So the total number of different orderings is n!/n = (n-1)!
Example: Flip a coin three times. S = ?
• S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
概率统计(英文) chapter 4 Continuous Random Variables and Probability Distributions
DEFINITION: Let X be a continuous rv. Then a probability distribution or probability density function of X is f(x) such that for any two numbers a and b with ab
4. Continuous Random Variables and Probability Distributions
4.1 Continuous Random Variables and Probability Density Functions
4.2 Cumulative Distribution Functions and Expected Values
•Example 4.3 :Suppose take a bus to work , and that every 5 minutes a bus arrives at my stop. Because of variation in the time that I leave my house , I don’t always arrive at the bus stop at the same time, so my waiting time X for the next bus is a continuous rv. The set of possible values of X is the interval [0,5]. One possible pdf for X is
4.3 The Normal Distribution
4.4 The Gamma Distribution and its Relatives
概率统计英文讲义Walpole02-23页精选文档
•AB=
The union of two events, A and B?
• The set of elements in either A or B. A B = ? • A B = {HHH, HHT, HTH, HTT , THT, TTH}.
Spring 2007
Event/Set Operations
Walpole Ch 02: 3
The complement of an event A?
• The set of all elements of S not in A. Denoted A’. • A = 1st flip is heads. A’ = first flip is not heads.
The intersection of two events A and B?
• The set of all elements in both A and B. Denoted A B. • B = 2nd or 3rd flip, but not both, are heads. • B = {HHT, HTH, THT, TTH}. A B = ? • A B = {HHT, HTH}
• What is a continuous vs. discrete sample space?
Example: Life in years of a component. S = ?
• S = {t | t 0} => “all values of t such that t 0” • A = component fails before the end of the fifth year. • A = {t | t < 5}.
概率论英文课件：ch1 Introduction
Topics (If time permits)1.Overview (Ch 1)2.Introduction to Probability (Ch 2 –Ch 7)3.From Probability to Inference : SamplingDistributions and Data Descriptions (Ch 8) 4.Estimation Process(Ch 9): One -and Two -Sample Point and Interval Estimate of aPopulation ParameterEvaluationThere will be weekly assignments:Assign on Wednesday Morning(Thursday Morning),while return on Monday Morning(Tuesday Afternoon)!The final grade of a student will be given based on his or her scores on the assignments(20%) and the final exam (80%).Correct use of language is one of the criteria included in the evaluation of all written work.IntroductionWhy Study Statistics and Probability?What Is Statistics?What is Probability?What Is the Role of Probability in Statistics?Happening i n Real Life1.Two football teams A and B have matchesin two sessions with score 4:1 and 5:0. Ifthey both met each other tomorrow, whichone could win?2.Two students promise to meet at the gate ofcampus library. The one who arrives firstwill wait the other for 15 minutes and willleave if waiting time span is bigger than 15 minutes. Will they meet successfully then?Happening i n Real Life3. Two students may have the same birthday ina class. What is the possibility ? This must happen for a class with a big enoughnumber.1-3*10-950.990.970.890.710.41905550403020Happening i n Real Life4. I can pass “the probability course”with theassurance of 80 percent.5. It is said that he has not fallen in love for fouryears in HUST. Right! Some boys complain: “Almost all of girls put their hears intobooks, not boys!”Right?6. It is reported that Chinese Yuan’s value isincreasing, while the value for Americandollar is decreasing more and more.Why Study Statistics and Probability •Statistics and probability is now an important part of everyday life, unavoidable and comes up in almost every walk of life.•Understanding the core ideas behind statistics and probability is crucial to your success in modern world.•By learning statistics and probability, you gain the competitive advantage of being comfortable and competent around data and uncertainty.Often we face to make decisions and create strategieswithout knowing the consequences with certainty.For examples:1. How many items of a particular product should be stocked in a large store that sells electronic products? (an inventory problem)2. Should a production line be shut down for checking malfunction if 5 items out of 100 are defective? (a quality control problem)3. Should a “new drug”for a certain disease be released for public (a medical research problem)4.What size of capacity should a university internet have in order to provide reliable, fast and inexpensive internet services? (queuing problem)5. Should you make an investment in a particular stock? (Investment problem)Probability and statistics can provide the models that help people make decisions such as these.Data collection Data analysis Model or Inference ApplicationWhy Study Statistics and Probability?All above examples involve a process of data collection, data summarization, data analysis and the decision making or prediction based on the data studied.Statistics and probability as decision-making tools play important roles in many areas of research and development and in prediction and control in a wide variety of fields.We often make decisions (prediction) or create strategies from knowledge, experience, and intuition. Statistics andprobability help you deal with the knowledge component,especially when this knowledge is in the form of numbers.Why Study Statistics and Probability?With rapid development of computer system and information technology, we are now living in the world filled with data.A vast amount of information is contained in data, but thisinformation is often not immediately accessible, statistics is helpful to extract and understand this information.Statistics and probability help you answer questions such as, To what extent should you really believe these figures and their implications? And, How should we summarize this mountain of data.For example, in our daily lives, we see that data are collected in stores, restaurants, hotels, and hospitals, …….What is statistics?Statistics is the science that deals with the collection, classification, analysis, and interpretation of information or data.By learning statistics, one gains the competitive advantage of being comfortable and competent around data and uncertainty.Descriptive statisticsDescriptive statistics includes the construction of graphs, charts, and tables, and the calculation of various descriptive numerical measures such as averages, measures of variation, and percentages.It utilizes numerical and graphical methods to summarize t he information revealed in a data set, and to present that information in a convenient form.Inferential statisticsInferential statistics utilizes sample data to make estimate, decision, prediction or other generalizations about a larger set (population) that contain the sample .PopulationThe collection of all individuals, items or data under consideration in a statistical study.SampleA subset selected from a population of interest.For example, in deciding on the choice of a job, you may wa nt to estimate the average salary being earned by people with degrees in the field.Population: All working people with degree in the field (or all salaries being earned by people with degree in field).Sample: A carefully selected group of working people with degree in the field.SampleSampleInferenceInferential StatisticsIn a statistical problem involving prediction or decision making, our primary interest is always in the population.In inferences basing on only a portion of population, we introduce an element of uncertainty. Statistical inference provides a measure of this uncertainty, and put a bound on the estimation or conclusion error.•An Overview on Statistics Method and Role of Probability Example 1.1 (Textbook):A manufacturer produces computer components.Since the production process is not perfect, it sometimes produces defective items.A quality control engineer wants to estimate the defect rate of theproduct, the percentage of the defective items in the entire production process.p = the defective rate of the product in the entire production process•Three Questions1. Is it possible or practical to find the exact value of p?2. How to estimate p?3. How accurate the estimate is?Steps to takeˆ()p2.Collect, organize and summarize the data from the sampleSuppose 100 items are randomly selected and of which 10 are defective.Estimate = 0.1 3.Find how good this estimation is (A single number estimate may not be precise)Is p in the interval [0.08, 0.12]?Cannot be sure of this since p is unknown and 0.1 is obtained based only on the small part of all the items produced.1.Sampling procedure (experimental design)How to choose a sample, how many to choose, what to measure and such?Inferential Statistic MethodWith statistic method, we can give an answer to the question: How confident or how sure we are that p is in the interval [0.08, 0.12]?For example, 95% certain that p is in the interval.How can we do that?Probability Theory !!!Probability studies the likelihood, or chances, for each of the various potential future events in a random process or a random phenomenon.Chance is all around us. Sometimes chance results from human design, as in the casino’s game of chance and statistician’s random samples. Sometimes nature use chance, as in choosing the sex of a child. Sometimes reason for chance behavior are mysterious, as when the number of deaths each year in a large population is as regular as the number of heads in many tosses of a coin.Even a brief introduction to probability will enable us to answer questions like these:How can gambling, which depends on the unpredictable fall of dice and cards, be a profitable business for a casino?Give a test for the ADS virus to the employees of a small company. What is the chance of at least one positive test if all the people tested are free of the virus?What the laws of probability says about the sex of a couple’s children? If you buy a lottery ticket every day for many years, how much will each ticket win on the average?Probability is the branch of mathematics that describes the pattern of chance outcomes. It helps us to understand risk and uncertainty.Using probability theory, we can build the mathematic model that describes the behaviors or patterns of outcomes of a random phenomenon or processes (For example, a queueing process, the behaviors of the inventory level of a product) Probability is an important field of mathematics that is rich in its depth of deductive reasoning and in its broad applications.Role of Probability in StatisticsRecall: In a statistical problem involving prediction or decision making, our primary interest is always inthe population.In inferences basing on only a portion of population, we introduce an element of uncertainty.With probability theory, Statistical inference provides a measure of this uncertainty, and put a bound on the estimation or conclusion error.So probability works together with statistics by providing a solid foundation for statistical inference.Role of Probability in StatisticsSo probability works together with statistics by providing a solid foundation for statistical inference.Probability is a what If tool for understanding risk and uncertainty.For example, you have an opportunity to invest in an oil exploration company. Past records show that out of ten previous oil drillings, all ten came up dry.What do you conclude?Similarly, if you toss a coin 20 times, and observe 19 “heads”and one “tail”, what can you conclude about the fairness of the coin.The Four Basic Activities of Statistics1)Design phaseIn the beginning stages of statistical study, either there are not yet any data or else it has not yet been decided what data to look closely at. Design phase will resolve these issues so that useful data will result.2) Exploratory phaseExploring the data involves looking at your data set from many angles, describing it, and summarizing it. It includes the construction of graphs, charts, and tables, and the calculation of various descriptive measures such as averages, measures of dispersion, and percentiles.The Four Basic Activities of Statistics3) Estimation ProcessA numerical summary of an unknown quantity, based on data, is the result the estimation process.We often need to estimate unknown quantities that are just plain impossible know exactly.Statistics can help us producing a good, educated estimate for an unknown quantity when reliable data are available.Statistical estimation also provides an indication of the amount of uncertainty error involved in the estimation.The Four Basic Activities of Statistics4) Hypothesis TestingStatistical hypothesis testing uses data to help us decide between two different possibilities in order to resolve an issue in an ambiguous situation.。
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Probability Intro
The sample space S of an experiment is the set of all possible outcomes.
• We must understand the sample space in order to determine the probability of each outcome occurring.
Venn Diagrams
Venn Diagrams show various events graphically, and are sometimes helpful in understanding set theory problems.
Standard set theory results hold:
• For three operations? • n1 n2 n3. • How many passwords of length 5 need to be checked by a
password hacking program if only lower case letters are used? • 265 = 11,881,376.
If one operation can be performed in n1 ways, and for each way, a second can be performed in n2 ways, then the two can be performed a total of n1 n2 ways.
• The set of all elements in both A and B. Denoted A B. • B = 2nd or 3rd flip, but not both, are heads. • B = {HHT, HTH, THT, TTH}. A B = ? • A B = {HHT, HTH}
• Why is this important? • This fact allows us to check our answers.
Properly enumerating the sample space is to correctly calculating probabilities.
• A tree diagram is sometimes useful.
Events
An event is a subset of a sample space (E S)
• Note that both S and are events as well.
Sample spaces can be continuous or discrete.
• What is a continuous vs. discrete sample space?
Example: Life in years of a component. S = ?
• S = {t | t 0} => “all values of t such that t 0” • A = component fails before the end of the fifth year. • A = {t | t < 5}.
Probability & Statistics for Engineers & Scientists, by
Walpole, Myers, Myers & Ye ~
Chapter 2 Notes
Class notes for ISE 201
San Jose State University Industrial & Systems Engineering Dept.
• The sample space is a set, the domain of the probability function.
• Each probability value, p, is a real number 0 p 1.
Important: The sum of the probabilities for all elements in the sample space always equals 1.
Two events are mutually exclusive if…?
•AB=
The union of two events, A and B?
• The set of elements in either A or B. A B = ? • A B = {HHH, HHT, HTH, HTT , THT, TTH}.
Example: Flip a coin three times. S = ?
• S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Event A = 1st flip is heads.
• A = {HHH, HHT, HTH, HTT}
Event/Set Operations
•A = ? •A = ? • A A’ = ? • A A’ = ? • S’ = ? • ’ = ? • (A’)’ = ? • (A B)’ = ? • (A B)’ = ? • (A B)’=A’ B’, (A B)’=A’ B’
Intuitive Sample Point Counting
The complement of an event A?
• The set of all elements of S not in A. Denoted A’. • A = 1st flip is heads. A’ = first flip is not heads.
The intersection of two events A and B?
This is called sampling with replacement.
A tree diagram can be used to enumerate all of the options.