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1、Random sampling (随机抽样) Independent（独立性） Representative（代表性） The political system of the USThe US Senate(参议院） Each state allows two senatorsSo there are about 100 senators in 50 states. The House of representatives（众议院） The number of representatives from each state depends on its population Californi
2、a 56 New York 33 Texas32 Delaware 3 Tennessee12 Florida25Using random numbers to implement random sampling128934567100Sampling probabilitySussex University has 1000 students. 200 of them are foreign students, others are American. What is the probability of randomly sampling a foreign student in one
3、trial? The sampling size If you select 1 student from 1000, then the outcome of probability would be either 0% or 100%. If you select many, then the outcome would approach 20%The coin-toss experimentExperimenterTimes of toss, nHead up times, mFrequency, m/nBuffon404020480.5069Pearson1200060190.5016P
4、earson24000120120.5005Head/TailThe law of large numbers(大数定律)When sampling size is sufficiently large, the sampling mean outcome calculated from repeated observations of a random variable must approach the distribution probability. limP|f/n p|=1nf : frequency of the eventsp : theoretical probability
5、n : sampling size : an infinitesimal numberThe law of large numbersThe law of large numbersis the foundation upon which such business enterprises as gambling casinos and insurance companies are builtSampling probabilitySussex University has 1000 students. 200 of them are foreign students, others are
6、 American. What is the probability of getting 2 foreign student in two samples? FFAAAFFASampling point (样本点）Sampling space（样本空间）FF, FA, AAp2, 2pq, q2p=PF = 200/1000q=PA = 800/1000p+q = 1The probability of independently sampling 2 studentsFFF, FFA, FAA, AAAp3, 3p2q, 3pq2, q3 The probability of indepe
7、ndently sampling 3 studentsGiven p=0.2, q=0.8, then the probability values:0.008, 0.096, 0.384 , 0.512Probability distribution (3 samples)00.20.40.6Numbers of American StudentsP0123Probability distributionFor samples of 1, p + qFor samples of 2, (p +q)2=p2 + 2pq + q2For samples of 3, (p +q)3=p3 + 3p
8、2q + 3pq2 + q3For samples of n, (p +q)n =The coefficients of the expanded terms Pascals triangle:1 11 2 11 3 3 11 4 6 4 11 5 10 10 5 1n12345For samples of 4, (p+q)4 =p4+4p3q + 6p2q2 + 4pq3 + q4The binomial distribution(二项分布)Pn(k) =Cnk pk qn-k X012knPqnnpqn-1Cn2 p2 qn-2Cnk pk qn-kpnDistribution serie
9、s (分布列） )!( !nNnNnNCnNCombinations ruleA sample of n elements to be chosen from a set of N elements. Then the number of different samples of n elements:Characteristics of the binomial distribution There are only two possible outcomes on each trial, F or A The trials are independent The probability o
10、f F (for a foreign student) remains the same from trial to trial, pThe binomial distribution(二项分布)Pn(k) =Cnk pk qn-k X012knPqnnpqn-1Cn2 p2 qn-2Cnk pk qn-kpnDistribution series (分布列）Property 1. 0 Pn(k) 1Property 2. Pn(k) = 1k=1n(p+q)n=1Basic probability rulesRule 1. Any probability P(A) is a number b
11、etween 0 and 1, that is 0 P(A) 1.Rule 2. The collection S of all possible outcomes has probability 1, that is P(S) =1.A discrete random variable X can assume five possible values: 2, 3, 5, 8, 10. Its probability distribution is shown here:X235810p(X)0.150.100.250.25a. What is p(5)?b. What is the pro
12、bability that X equals 2 or 10c. What is p(X8)?The mean, expected value, or expectationof a discrete variable x is defined as = E(x) = xp(x)What is the average size of an American family? Here is the distribution of the size of American families according to Census Bureau studies.Persons in family23
13、45678Fraction of families.396 .231 .212 .096.038.017 .008Random discrete variables(离散型随机变量) The law of large numbers The binomial distribution 甲、已两台车床加工同一型号产品，生产1000只所含次品数各用X, Y表示，经过一端时间观察，X, Y的分布列各为X0123P0.70.10.10.1Y0123P0.50.30.20Which machine has a better quality?Homework1.Page 45, question 3.6
14、2.Please write out a macro program for generating 200 random numbers in the range from 0 to 100 and storing the data in the first two columns on Excel spreadsheet. 3.In human beings the sex ratio of newborn infants is about 100 male to 105 female. Were we to take 10,000 random samples of 6 newborn infants from the total population of such infants for one year, what would be the expected frequency of groups of 6 males, 5 males, 4 males, and so on?