3 6 Chapter Discrete Probability Distributions

3 6 Chapter Discrete Probability Distributions
© 2010 Pearson Prentice Hall. All rights reserved

Section 6.1 Probability Rules

A random variable is a numerical measure of the outcome from a probability experiment, so its value is determined by chance. Random variables are denoted using letters such as X. © 2010 Pearson Prentice Hall. All rights reserved

A discrete random variable has either a finite or countable number of values. The values of a discrete random variable can be plotted on a number line with space between each point. See the figure. © 2010 Pearson Prentice Hall. All rights reserved

A continuous random variable has infinitely many values. The values of a continuous random variable can be plotted on a line in an uninterrupted fashion. See the figure. © 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE Distinguishing Between Discrete and Continuous Random Variables Determine whether the following random variables are discrete or continuous. State possible values for the random variable. The number of light bulbs that burn out in a room of 10 light bulbs in the next year. (b) The number of leaves on a randomly selected Oak tree. (c) The length of time between calls to 911. Discrete; x = 0, 1, 2, …, 10 Discrete; x = 0, 1, 2, … Continuous; t > 0 © 2010 Pearson Prentice Hall. All rights reserved

A probability distribution provides the possible values of the random variable X and their corresponding probabilities. A probability distribution can be in the form of a table, graph or mathematical formula. © 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE A Discrete Probability Distribution The table to the right shows the probability distribution for the random variable X, where X represents the number of DVDs a person rents from a video store during a single visit. x P(x) 0.06 1 0.58 2 0.22 3 0.10 4 0.03 5 0.01 © 2010 Pearson Prentice Hall. All rights reserved

A probability histogram is a histogram in which the horizontal axis corresponds to the value of the random variable and the vertical axis represents the probability of that value of the random variable. © 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE Drawing a Probability Histogram x P(x) 0.06 1 0.58 2 0.22 3 0.10 4 0.03 5 0.01 Draw a probability histogram of the probability distribution to the right, which represents the number of DVDs a person rents from a video store during a single visit. © 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE Computing the Mean of a Discrete Random Variable Compute the mean of the probability distribution to the right, which represents the number of DVDs a person rents from a video store during a single visit. x P(x) 0.06 1 0.58 2 0.22 3 0.10 4 0.03 5 0.01 © 2010 Pearson Prentice Hall. All rights reserved

Because the mean of a random variable represents what we would expect to happen in the long run, it is also called the expected value, E(X), of the random variable. © 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE Computing the Expected Value of a Discrete Random Variable A term life insurance policy will pay a beneficiary a certain sum of money upon the death of the policy holder. These policies have premiums that must be paid annually. Suppose a life insurance company sells a $250,000 one year term life insurance policy to a 49-year-old female for $530. According to the National Vital Statistics Report, Vol. 47, No. 28, the probability the female will survive the year is Compute the expected value of this policy to the insurance company. x P(x) 530 530 – 250,000 = -249,470 Survives Does not survive E(X) = 530( ) + (-249,470)( ) = $7.50 © 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE Computing the Variance and Standard Deviation of a Discrete Random Variable x P(x) 0.06 1 0.58 2 0.22 3 0.10 4 0.03 5 0.01 Compute the variance and standard deviation of the following probability distribution which represents the number of DVDs a person rents from a video store during a single visit. x P(x) 0.06 -1.43 2.0449 1 0.58 -0.91 0.8281 2 0.22 -1.27 1.6129 3 0.1 -1.39 1.9321 4 0.03 -1.46 2.1316 5 0.01 -1.48 2.1904 © 2010 Pearson Prentice Hall. All rights reserved

Section 6.2 The Binomial Probability Distribution

Criteria for a Binomial Probability Experiment An experiment is said to be a binomial experiment if 1. The experiment is performed a fixed number of times. Each repetition of the experiment is called a trial. 2. The trials are independent. This means the outcome of one trial will not affect the outcome of the other trials. 3. For each trial, there are two mutually exclusive (or disjoint) outcomes, success or failure. 4. The probability of success is fixed for each trial of the experiment. © 2010 Pearson Prentice Hall. All rights reserved

Notation Used in the Binomial Probability Distribution
There are n independent trials of the experiment. Let p denote the probability of success so that 1 – p is the probability of failure. Let X be a binomial random variable that denotes the number of successes in n independent trials of the experiment. So, 0 < x < n. © 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE Identifying Binomial Experiments Which of the following are binomial experiments? A player rolls a pair of fair die 10 times. The number X of 7’s rolled is recorded. (b) The 11 largest airlines had an on-time percentage of 84.7% in November, 2001 according to the Air Travel Consumer Report. In order to assess reasons for delays, an official with the FAA randomly selects flights until she finds 10 that were not on time. The number of flights X that need to be selected is recorded. (c) In a class of 30 students, 55% are female. The instructor randomly selects 4 students. The number X of females selected is recorded. Binomial experiment Not a binomial experiment – not a fixed number of trials. Not a binomial experiment – the trials are not independent. © 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE Constructing a Binomial Probability Distribution According to the Air Travel Consumer Report, the 11 largest air carriers had an on-time percentage of 79.0% in May, Suppose that 4 flights are randomly selected from May, 2008 and the number of on-time flights X is recorded. Construct a probability distribution for the random variable X using a tree diagram. © 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE Using the Binomial Probability Distribution Function According to the Experian Automotive, 35% of all car-owning households have three or more cars. In a random sample of 20 car-owning households, what is the probability that exactly 5 have three or more cars? (b) In a random sample of 20 car-owning households, what is the probability that less than 4 have three or more cars? (c) In a random sample of 20 car-owning households, what is the probability that at least 4 have three or more cars? © 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE Finding the Mean and Standard Deviation of a Binomial Random Variable According to the Experian Automotive, 35% of all car-owning households have three or more cars. In a simple random sample of 400 car-owning households, determine the mean and standard deviation number of car-owning households that will have three or more cars. © 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE Constructing Binomial Probability Histograms (a) Construct a binomial probability histogram with n = 8 and p = 0.15. (b) Construct a binomial probability histogram with n = 8 and p = 0. 5. (c) Construct a binomial probability histogram with n = 8 and p = 0.85. For each histogram, comment on the shape of the distribution. © 2010 Pearson Prentice Hall. All rights reserved

For a fixed probability of success, p, as the number of trials n in a binomial experiment increase, the probability distribution of the random variable X becomes bell-shaped. As a general rule of thumb, if np(1 – p) > 10, then the probability distribution will be approximately bell-shaped. © 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE Using the Mean, Standard Deviation and Empirical Rule to Check for Unusual Results in a Binomial Experiment According to the Experian Automotive, 35% of all car-owning households have three or more cars. A researcher believes this percentage is higher than the percentage reported by Experian Automotive. He conducts a simple random sample of 400 car-owning households and found that 162 had three or more cars. Is this result unusual ? The result is unusual since 162 > 159.1 © 2010 Pearson Prentice Hall. All rights reserved

Section 6.3 The Poisson Probability Distribution

A random variable X, the number of successes in a fixed interval, follows a Poisson process provided the following conditions are met 1. The probability of two or more successes in any sufficiently small subinterval is 0. 2. The probability of success is the same for any two intervals of equal length. 3. The number of successes in any interval is independent of the number of successes in any other interval provided the intervals are not overlapping. © 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE Illustrating a Poisson Process The Food and Drug Administration sets a Food Defect Action Level (FDAL) for various foreign substances that inevitably end up in the food we eat and liquids we drink. For example, the FDAL level for insect filth in chocolate is 0.6 insect fragments (larvae, eggs, body parts, and so on) per 1 gram. © 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE Illustrating a Poisson Process The Food and Drug Administration sets a Food Defect Action Level (FDAL) for various foreign substances that inevitably end up in the food we eat and liquids we drink. For example, the FDAL level for insect filth in chocolate is 0.6 insect fragments (larvae, eggs, body parts, and so on) per 1 gram. Suppose that a chocolate bar has 0.6 insect fragments per gram. Compute the probability that the number of insect fragments in a 10-gram sample of chocolate is exactly three. Interpret the result. fewer than three. Interpret the result. at least three. Interpret the result. © 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE Mean and Standard Deviation of a Poisson Random Variable The Food and Drug Administration sets a Food Defect Action Level (FDAL) for various foreign substances that inevitably end up in the food we eat and liquids we drink. For example, the FDAL level for insect filth in chocolate is 0.6 insect fragments (larvae, eggs, body parts, and so on) per 1 gram. Determine the mean number of insect fragments in a 5 gram sample of chocolate. What is the standard deviation? (a) (b) We would expect 3 insect fragments in a 5-gram sample of chocolate. © 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE A Poisson Process? In 1910, Ernest Rutherford and Hans Geiger recorded the number of -particles emitted from a polonium source in eighth-minute (7.5 second) intervals. The results are reported in the table to the right. Does a Poisson probability function accurately describe the number of -particles emitted? Source: Rutherford, Sir Ernest; Chadwick, James; and Ellis, C.D.. Radiations from Radioactive Substances. London, Cambridge University Press, 1951, p. 172. © 2010 Pearson Prentice Hall. All rights reserved

Section 6.4 The Hypergeometric Probability Distribution

EXAMPLE A Hypergeometric Probability Distribution The Dow Jones Industrial Average (DJIA) is a collection of thirty publicly traded companies that are meant to be representative of the United States economy. In one certain month 18 of the 30 stocks in the DJIA increased in value. If an investor randomly invests in four stocks at the beginning of this month and records X, the number of stocks that increased in value during the month, is this a hypergeometric probability experiment? List the possible values of the random variable X. This is a hypergeometric probability experiment because The population consists of N = 30 stocks. Two outcomes are possible: either the stock increases in value, or it does not increase in value. There are k = 18 successes. The sample size is n = 4. The possible values of the random variable are x = 0, 1, 2, 3, 4. © 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE A Hypergeometric Probability Distribution The Dow Jones Industrial Average (DJIA) is a collection of thirty publicly traded companies that are meant to be representative of the United States economy. In one certain month 18 of the 30 stocks in the DJIA increased in value. What is the probability that an investor randomly invests in four stocks at the beginning of this month and three of the stocks increased in value? N = 30; n = 4; k = 18; x = 3 © 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE A Hypergeometric Probability Distribution The Dow Jones Industrial Average (DJIA) is a collection of thirty publicly traded companies that are meant to be representative of the United States economy. In one certain month 18 of the 30 stocks in the DJIA increased in value. In a random sample of four stocks, determine the mean and standard deviation of the number of stocks that will increase in value. N = 30; n = 4; k = 18 We would expect 2.4 stocks out of 4 stocks to increase in value during the month. © 2010 Pearson Prentice Hall. All rights reserved

3 6 Chapter Discrete Probability Distributions

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