《統(tǒng)計學(xué)基礎(chǔ)(英文版·第7版)》課件les7e-ADA-0401

1、Elementary StatisticsSeventh EditionChapter 4Discrete Probability DistributionsCopyright 2019, 2015, 2012, Pearson Education, Inc.Elementary StatisticsSeventh EChapter Outline4.1 Probability Distributions4.2 Binomial Distributions4.3 More Discrete Probability DistributionsChapter Outline4.1 Probabil

2、itySection 4.1Probability DistributionsSection 4.1Probability DistribSection 4.1 ObjectivesHow to distinguish between discrete random variables and continuous random variablesHow to construct a discrete probability distribution and its graph and how to determine if a distribution is a probability di

3、stributionHow to find the mean, variance, and standard deviation of a discrete probability distributionHow to find the expected value of a discrete probability distributionSection 4.1 ObjectivesHow to dRandom Variables (1 of 3)Random VariableRepresents a numerical value associated with each outcome

4、of a probability distribution.Denoted by xExamples . .Random Variables (1 of 3)RandoRandom Variables (2 of 3)Discrete Random VariableHas a finite or countable number of possible outcomes that can be listed.Example .Random Variables (2 of 3)DiscrRandom Variables (3 of 3)Continuous Random VariableHas

5、an uncountable number of possible outcomes, represented by an interval on the number line.Example .Random Variables (3 of 3)ContiExample: Discrete and Continuous Variables (1 of 2)Determine whether each random variable x is discrete or continuous. Explain your reasoning.Let x represent the number of

6、 Fortune 500 companies that lost money in the previous year.Solution:Discrete random variable (The number of companies that lost money in the previous year can be counted.)Example: Discrete and ContinuoExample: Discrete and Continuous Variables (2 of 2)Determine whether each random variable x is dis

7、crete or continuous. Explain your reasoning.Let x represent the volume of gasoline in a 21-gallon tank.Solution:Continuous random variable (The amount of gasoline in the tank can be any volume between 0 gallons and 21 gallons.)Example: Discrete and ContinuoDiscrete Probability DistributionsDiscrete

8、probability distributionLists each possible value the random variable can assume, together with its probability. Must satisfy the following conditions:In WordsIn SymbolsThe probability of each value of the discrete random variable is between 0 and 1, inclusive.The sum of all the probabilities is 1.D

9、iscrete Probability DistributConstructing a Discrete Probability DistributionLet x be a discrete random variable with possible outcomes .Make a frequency distribution for the possible outcomes.Find the sum of the frequencies.Find the probability of each possible outcome by dividing its frequency by

10、the sum of the frequencies.Check that each probability is between 0 and 1, inclusive, and that the sum of all the probabilities is 1.Constructing a Discrete ProbabExample: Constructing and Graphing a Discrete Probability DistributionAn industrial psychologist administered a personality inventory tes

11、t for passive-aggressive traits to 150 employees. Each individual was given a whole number score from 1 to 5, where 1 is extremely passive and 5 is extremely aggressive. A score of 3 indicated neither trait. The results are shown. Construct a probability distribution for the random variable x. Then

12、graph the distribution using a histogram.Score, xFrequency, f124233342430521Example: Constructing and GrapSolution: Constructing and Graphing a Discrete Probability Distribution (1 of 3)Divide the frequency of each score by the total number of individuals in the study to find the probability for eac

13、h value of the random variable.Discrete probability distribution: x12345P(x)0.160.220.280.200.14Solution: Constructing and GraSolution: Constructing and Graphing a Discrete Probability Distribution (2 of 3) x12345P(x)0.160.220.280.200.14This is a valid discrete probability distribution since Each pr

14、obability is between 0 and 1, inclusive, .The sum of the probabilities equals 1, .Solution: Constructing and GraSolution: Constructing and Graphing a Discrete Probability Distribution (3 of 3) x12345P(x)0.160.220.280.200.14Because the width of each bar is one, the area of each bar is equal to the pr

15、obability of a particular outcome. Also, the probability of an event corresponds to the sum of the areas of the outcomes included in the event.You can see that the distribution is approximately symmetric.Passive-Aggressive TraitsSolution: Constructing and GraExample: Verifying a Probability Distribu

16、tionVerify that the distribution for the three-day forecast and the number of days of rain is a probability distribution.Days of Rain, x0123Probability, P(x)0.2160.4320.2880.064Example: Verifying a ProbabiliSolution: Verifying a Probability DistributionSolutionIf the distribution is a probability di

17、stribution, then (1) each probability is between 0 and 1, inclusive, and (2) the sum of all the probabilities equals 1.Each probability is between 0 and 1. .Days of Rain, x0123Probability, P(x)0.2160.4320.2880.064Because both conditions are met, the distribution is a probability distribution.Solutio

18、n: Verifying a ProbabilExample: Identifying Probability Distributions (1 of 2)Determine whether each distribution is a probability distribution. Explain your reasoning. x5678P(x)0.280.210.430.15SolutionEach probability is between 0 and 1, but the sum of all the probabilities is 1.07, which is greate

19、r than 1. The sum of all the probabilities in a probability distribution always equals 1. So, this distribution is not a probability distribution.Example: Identifying ProbabiliExample: Identifying Probability Distributions (2 of 2)Determine whether each distribution is a probability distribution. Ex

20、plain your reasoning. SolutionThe sum of all the probabilities is equal to 1, but P(3) and P(4) are not between 0 and 1. Probabilities can never be negative or greater than 1. So, this distribution is not a probability distribution.Example: Identifying ProbabiliMeanMean of a discrete probability dis

21、tribution Each value of x is multiplied by its corresponding probability and the products are added.MeanMean of a discrete probabiExample: Finding the MeanThe probability distribution for the personality inventory test for passive-aggressive traits is given. Find the mean score.Solution:Example: Fin

22、ding the MeanThe pSolution: Finding the MeanThe probability distribution for the personality inventory test for passive-aggressive traits is given. Find the mean score.Solution:Recall that a score of 3 represents an individual who exhibits neither passive nor aggressive traits and the mean is slight

23、ly less than 3. So, the mean personality trait is neither extremely passive nor extremely aggressive, but is slightly closer to passive.Solution: Finding the MeanThe Variance and Standard DeviationVariance of a discrete probability distribution Standard deviation of a discrete probability distributi

24、on Variance and Standard DeviatioExample: Finding the Variance and Standard DeviationThe probability distribution for the personality inventory test for passive-aggressive traits is given. Find the variance and standard deviation.Score, xProbability, P(x)10.1620.2230.2840.2050.14Example: Finding the Variance Solution: Finding the Variance and Standard DeviationRecall Variance: Standard Deviation: Most of the data values differ from the mean by no more than 1.3.Solution: Finding the VarianceExpected