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1、Elementary Statistics,Seventh Edition,Chapter 4,Discrete Probability Distributions,Copyright 2019, 2015, 2012, Pearson Education, Inc.,Elementary StatisticsSeventh E,Chapter Outline,4.1 Probability Distributions4.2 Binomial Distributions4.3 More Discrete Probability Distributions,Chapter Outline4.1
2、Probability,Section 4.1,Probability Distributions,Section 4.1Probability Distrib,Section 4.1 Objectives,How 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 p
3、robability distributionHow to find the mean, variance, and standard deviation of a discrete probability distributionHow to find the expected value of a discrete probability distribution,Section 4.1 ObjectivesHow to d,Random Variables (1 of 3),Random VariableRepresents a numerical value associated wi
4、th each outcome of a probability distribution.Denoted by xExamples,.,.,Random Variables (1 of 3)Rando,Random 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)Discr,Random Variables (3 of 3),Continuo
5、us Random VariableHas an uncountable number of possible outcomes, represented by an interval on the number line.Example,.,Random Variables (3 of 3)Conti,Example: Discrete and Continuous Variables (1 of 2),Determine whether each random variable x is discrete or continuous. Explain your reasoning.,Let
6、 x represent the number of 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 Continuo,Example: Discrete and Continuous Variables (2 of 2),Determine whether
7、 each random variable x is discrete 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 Continuo,Discrete
8、Probability Distributions,Discrete probability distributionLists each possible value the random variable can assume, together with its probability. Must satisfy the following conditions:,In Words,In Symbols,The probability of each value of the discrete random variable is between 0 and 1, inclusive.,
9、The sum of all the probabilities is 1.,Discrete Probability Distribut,Constructing a Discrete Probability Distribution,Let 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 pos
10、sible outcome by dividing its frequency by 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 Probab,Example: Constructing and Graphing a Discrete Probability Distribution,An industrial psychol
11、ogist administered a personality inventory test 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
12、 distribution for the random variable x. Then graph the distribution using a histogram.,Example: Constructing and Grap,Solution: 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 proba
13、bility for each value of the random variable.,Discrete probability distribution:,Solution: Constructing and Gra,Solution: Constructing and Graphing a Discrete Probability Distribution (2 of 3),This is a valid discrete probability distribution since Each probability is between 0 and 1, inclusive,.,Th
14、e sum of the probabilities equals 1,.,Solution: Constructing and Gra,Solution: Constructing and Graphing a Discrete Probability Distribution (3 of 3),Because the width of each bar is one, the area of each bar is equal to the probability of a particular outcome. Also, the probability of an event corr
15、esponds to the sum of the areas of the outcomes included in the event.,You can see that the distribution is approximately symmetric.,Passive-Aggressive Traits,Solution: Constructing and Gra,Example: Verifying a Probability Distribution,Verify that the distribution for the three-day forecast and the
16、number of days of rain is a probability distribution.,Example: Verifying a Probabili,Solution: Verifying a Probability Distribution,SolutionIf the distribution is a probability distribution, then (1) each probability is between 0 and 1, inclusive, and (2) the sum of all the probabilities equals 1.,E
17、ach probability is between 0 and 1.,.,Because both conditions are met, the distribution is a probability distribution.,Solution: Verifying a Probabil,Example: Identifying Probability Distributions (1 of 2),Determine whether each distribution is a probability distribution. Explain your reasoning.,Sol
18、utionEach probability is between 0 and 1, but the sum of all the probabilities is 1.07, which is greater 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 Probabili,Example: Identifyin
19、g Probability Distributions (2 of 2),Determine whether each distribution is a probability distribution. Explain 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 distribut
20、ion is not a probability distribution.,Example: Identifying Probabili,Mean,Mean of a discrete probability distribution,Each value of x is multiplied by its corresponding probability and the products are added.,MeanMean of a discrete probabi,Example: Finding the Mean,The probability distribution for
21、the personality inventory test for passive-aggressive traits is given. Find the mean score.,Solution:,Example: Finding the MeanThe p,Solution: Finding the Mean,The probability distribution for the personality inventory test for passive-aggressive traits is given. Find the mean score.,Solution:,Recal
22、l that a score of 3 represents an individual who exhibits neither passive nor aggressive traits and the mean is slightly 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 St
23、andard Deviation,Variance of a discrete probability distribution,Standard deviation of a discrete probability distribution,Variance and Standard Deviatio,Example: Finding the Variance and Standard Deviation,The probability distribution for the personality inventory test for passive-aggressive traits
24、 is given. Find the variance and standard deviation.,Example: Finding the Variance,Solution: Finding the Variance and Standard Deviation,Recall,Variance:,Standard Deviation:,Most of the data values differ from the mean by no more than 1.3.,Solution: Finding the Variance,Expected Value,Expected value
25、 of a discrete random variable Equal to the mean of the random variable.,Expected ValueExpected value o,Example: Finding an Expected Value,At a raffle, 1500 tickets are sold at,each for four,prizes of, and,. You buy one,ticket. Find the expected value and interpret its meaning.,Example: Finding an E
26、xpected V,Solution: Finding an Expected Value (1 of 2),To find the gain for each prize, subtract the price of the ticket from the prize:,Your gain for the,prize is,Your gain for the,prize is,Your gain for the,prize is,Your gain for the,prize is,If you do not win a prize, your gain is,Solution: Finding an Expected,Solution: Finding an Expected Value (2 of 2),Probability distribution for the possible gains (outcomes),You can expect to lose an average of,for each,ticket you buy.,Solution: Finding an Expected,
