经管学会网盘概统lecture

1、Lecture 12The Normal DistributionThe Central Limit TheoremThe Normal Distribution The normal distribution is the single most important probability distribution.Continuous DistributionMany real -life data sets look like this one, the name given to this general shape is “normal” Normal distributionImp

2、ortance of Normal Distribution1. Describes Many Random Processes or Continuous Phenomena2. The central limit theorem. If a large random sample is taken from some distribution, then even though this distribution is not itself approximately normal, many important functions of the observations in the s

3、ample will have distributions which are approximately normal.3. Basis for Classical Statistical InferenceDefinition of the Normal DistributionA random variable X has a normal distribution with mean and varianceif X has a continuous distribution with p.d.f. The Shape of the Normal DistributionThe p.d

4、.f. of a normal distribution is symmetric with respect to the point x=m. Linear TransformationTheorem . If X has a normal distribution with mean and variance and if Y=aX+b, where a and b are given constants and , then Y has a normal distribution with mean and variance .The Standard Normal Distributi

5、onThe normal distribution with mean 0 and variance 1 is called the standard normal distribution. The p.d.f. of Z that follows the standard normal distribution is denoted by the symbol , and the d.f. is denoted by the symbol .Normal Distribution ProbabilityProbability is area under curve!Infinite Num

6、ber of TablesNormal distributions differ by mean & standard deviation.Each distribution would require its own table.Thats an infinite number!Standardize the Normal Distribution One table!Normal DistributionStandardized Normal DistributionStandardizing ExampleNormal DistributionStandardized Normal Di

7、stributionNormal Probability TablesExample: P(Z 2.00) = .9773 The Standardized Normal table in the textbook (Appendix) gives the value of for Z02.00.9773Notice thatSo values of can be derived for z0. If a random variable X has a normal distribution with mean and variance , then the variable has a st

8、andard normal distribution. So probabilities for any normal distribution can be derived.Normal Distribution Thinking ChallengeYou work in Quality Control for GE. Light bulb life has a normal distribution with = 2000 hours & = 200 hours. Whats the probability that a bulb will lastA. between 2000 & 24

9、00 hours?B. less than 1470 hours?Solution* P(2000 X 2400)Normal Distribution .4773Standardized Normal DistributionSolution* P(X 1470)Normal Distribution.4960 .0040.5000Standardized Normal DistributionFinding X Values for Known ProbabilitiesNormal DistributionStandardized Normal Distribution .1217 .1

10、217Shaded areas exaggeratedProperties of the Normal DistributionThe area under the part of a normal curve that lies within 1standard deviation of the mean is approximately 0.68, or 68%;within 2 standard deviations, about 0.95, or 95%; and within 3standard deviations, about 0.997, or 99.7%.Linear Com

11、binations of Normally Distributed VariablesTheorem. If the random variables X1,.,Xk are independent and if Xi has a normal distribution with mean and variance (i=1,.,k), then the sum X1+.+Xk has a normal distribution with mean and variance .Corollary 1. If the random variables X1,.,Xk are independen

12、t, if Xi has a normal distribution with mean and variance (i=1,.,k), and if a1,.,ak and b are constants for which at least one of the values a1,.,ak is different from 0, then the variable a1X1+.+akXk+b has a normal distribution with mean and variance .Corollary 2. Suppose that the random variables X

13、1,.,Xn form a random sample from a normal distribution with and variance , and let denote the sample mean. Then has a normal distribution with mean and variance .ExampleSuppose that the heights, in inches, of the women in a certain population follow a normal distribution with mean 65 and standard de

14、viation 1, and that the heights of the men follow a normal distribution with mean 68 and standard deviation 2. Suppose that one woman is selected at random, and independently, one man is selected at random. What is the probability that the woman will be taller than the man?Solution: Let W denote the

15、 height of the selected woman, and let M denote the height of the selected man. Then the difference W-M has a normal distribution with mean 65-68=-3 and variance LetThen Z has a standard normal distribution. SoExample: Determining a Sample SizeSuppose that a random sample of size n is to be taken fr

16、om a normal distribution with mean and variance 9. What is the miminum value of n for which Solution: The sample mean has a normal distribution with mean and standard deviationLet , then Z has a standard normal distribution, and The sample size must be at least 35.28统计分析的任务通过样本的统计量来了解总体的参数。总体参数p样本统计

17、量为什么需要抽样？ 1）总体无法得到。例：光临麦当劳的所有顾客（无限总体）。 2）时间和成本不允许。例：美国总统选举的民意测验。 3）实验具有破坏性。例：测量产品的寿命。抽取的样本不同，那么算出的平均值也不同 抽样分布抽取的样本不同，那么算出的平均值也不同。需要了解样本平均值的分布，即它的抽样分布。样本均值的抽样分布样本均值的抽样分布计算出每个样本的均值，如下表。并给出样本均值的抽样分布样本均值的分布和总体的分布36关于抽样分布的神奇现象对于简单随机抽样不管总体的分布是什么形态，设它的均值是，方差是2。只要样本的容量n很大，那么样本的均值总是近似服从正态分布(中心极限定理)If a large

18、 random sample is taken from any distribution with mean and variance ,regardless of the distributional form, The distribution of the sum will be approximately a normal distribution with mean and variance .The Central Limit TheoremExample: Tossing a CoinSuppose a fair coin is tossed 900 times. What is the probability of obtaining more than 495 heads?For i=1,.,900, let Xi=1 if a head is obtained on the ith toss and let Xi=0 otherwise. Then E(Xi)=1/2 and Var(Xi)=1/4. From the central limit theorem, the total number of heads will be approximately a normal distribution with mean (900)(1/2)=45