STATISTICS INFORMED DECISIONS USING DATA Fifth Edition Chapter 6 Discrete Probability Distributions Copyright © 2017, 2013, 2010 Pearson Education, Inc. All Rights Reserved

6.1 Discrete Random Variables Learning Objectives Distinguish between discrete and continuous random variables Identify discrete probability distributions Graph discrete probability distributions Compute and interpret the mean of a discrete random variable Interpret the mean of a discrete random variable as an expected value Compute the standard deviation of a discrete random variable

6. 1 Discrete Random Variables 6. 1 6.1 Discrete Random Variables 6.1.1 Distinguish between Discrete and Continuous Random Variables (1 of 4) 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.

6. 1 Discrete Random Variables 6. 1 6.1 Discrete Random Variables 6.1.1 Distinguish between Discrete and Continuous Random Variables (2 of 4) 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.

6. 1 Discrete Random Variables 6. 1 6.1 Discrete Random Variables 6.1.1 Distinguish between Discrete and Continuous Random Variables (3 of 4) 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.

6. 1 Discrete Random Variables 6. 1 6.1 Discrete Random Variables 6.1.1 Distinguish between Discrete and Continuous Random Variables (4 of 4) 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. (a) The number of light bulbs that burn out in a room of 10 light bulbs in the next year. Discrete; x = 0, 1, 2, …, 10 (b) The number of leaves on a randomly selected oak tree. Discrete; x = 0, 1, 2, … (c) The length of time between calls to 911. Continuous; t > 0

6. 1 Discrete Random Variables 6. 1 6.1 Discrete Random Variables 6.1.2 Identify Discrete Probability Distributions (1 of 6) The probability distribution of a discrete random variable X provides the possible values of the random variable and their corresponding probabilities. A probability distribution can be in the form of a table, graph or mathematical formula.

6. 1 Discrete Random Variables 6. 1 6.1 Discrete Random Variables 6.1.2 Identify Discrete Probability Distributions (2 of 6) 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 movies streamed on Netflix each month. x P(x) 0.06 1 0.58 2 0.22 3 0.10 4 0.03 5 0.01

6. 1 Discrete Random Variables 6. 1 6.1 Discrete Random Variables 6.1.2 Identify Discrete Probability Distributions (3 of 6) Rules for a Discrete Probability Distribution Let P(x) denote the probability that the random variable X equals x; then ΣP(x) = 1 0 ≤ P(x) ≤ 1

6. 1 Discrete Random Variables 6. 1 6.1 Discrete Random Variables 6.1.2 Identify Discrete Probability Distributions (4 of 6) EXAMPLE Identifying Probability Distributions Is the following a probability distribution? x P(x) 0.16 1 0.18 2 0.22 3 0.10 4 0.30 5 0.01

6. 1 Discrete Random Variables 6. 1 6.1 Discrete Random Variables 6.1.2 Identify Discrete Probability Distributions (5 of 6) EXAMPLE Identifying Probability Distributions Is the following a probability distribution? x P(x) 0.16 1 0.18 2 0.22 3 0.10 4 0.30 5 −0.01

6. 1 Discrete Random Variables 6. 1 6.1 Discrete Random Variables 6.1.2 Identify Discrete Probability Distributions (6 of 6) EXAMPLE Identifying Probability Distributions Is the following a probability distribution? x P(x) 0.16 1 0.18 2 0.22 3 0.10 4 0.30 5 0.04

6. 1 Discrete Random Variables 6. 1 6.1 Discrete Random Variables 6.1.3 Graph Discrete Probability Distributions EXAMPLE Drawing a Probability Histogram Draw a probability histogram of the probability distribution to the right, which represents the number of movies streamed on Netflix each month. x P(x) 0.06 1 0.58 2 0.22 3 0.10 4 0.03 5 0.01

6. 1 Discrete Random Variables 6. 1 6.1 Discrete Random Variables 6.1.4 Compute and Interpret the Mean of a Discrete Random Variable (1 of 6) The Mean of a Discrete Random Variable The mean of a discrete random variable is given by the formula

6. 1 Discrete Random Variables 6. 1 6.1 Discrete Random Variables 6.1.4 Compute and Interpret the Mean of a Discrete Random Variable (2 of 6) 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

6. 1 Discrete Random Variables 6. 1 6.1 Discrete Random Variables 6.1.4 Compute and Interpret the Mean of a Discrete Random Variable (3 of 6) Interpretation of the Mean of a Discrete Random Variable Suppose an experiment is repeated n independent times and the value of the random variable X is recorded. As the number of repetitions of the experiment increases, the mean value of the n trials will approach μX, the mean of the random variable X. In other words, let x1 be the value of the random variable X after the first experiment, x2 be the value of the random variable X after the second experiment, and so on. Then

6. 1 Discrete Random Variables 6. 1 6.1 Discrete Random Variables 6.1.4 Compute and Interpret the Mean of a Discrete Random Variable (4 of 6) EXAMPLE Interpretation of the Mean of a Discrete Random Variable The following data represent the number of DVDs rented by 100 randomly selected customers in a single visit. Compute the mean number of DVDs rented. 1 2 3 4 5

6. 1 Discrete Random Variables 6. 1 6.1 Discrete Random Variables 6.1.4 Compute and Interpret the Mean of a Discrete Random Variable (5 of 6)

6. 1 Discrete Random Variables 6. 1 6.1 Discrete Random Variables 6.1.4 Compute and Interpret the Mean of a Discrete Random Variable (6 of 6) As the number of trials of the experiment increases, the mean number of rentals approaches the mean of the probability distribution.

6. 1 Discrete Random Variables 6. 1 6.1 Discrete Random Variables 6.1.5 Compute and Interpret the Mean of a Discrete Random Variable (1 of 2) 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.

6. 1 Discrete Random Variables 6. 1 6.1 Discrete Random Variables 6.1.5 Compute and Interpret the Mean of a Discrete Random Variable (2 of 2) 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 0.99791. Compute the expected value of this policy to the insurance company. blank x P(x) Survives 530 0.99791 Does not survive 530 − 250,000 = −249,470 0.00209

6. 1 Discrete Random Variables 6. 1 6.1 Discrete Random Variables 6.1.5 Compute the Standard Deviation of a Discrete Random Variable (1 of 3) Standard Deviation of a Discrete Random Variable The standard deviation of a discrete random variable X is given by

6. 1 Discrete Random Variables 6. 1 6.1 Discrete Random Variables 6.1.5 Compute the Standard Deviation of a Discrete Random Variable (2 of 3) EXAMPLE Computing the Variance and Standard Deviation of a Discrete Random Variable 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. Remember, the mean that we found was 1.49. x P(x) 0.06 1 0.58 2 0.22 3 0.10 4 0.03 5 0.01

6. 1 Discrete Random Variables 6. 1 6.1 Discrete Random Variables 6.1.5 Compute the Standard Deviation of a Discrete Random Variable (3 of 3) EXAMPLE Computing the Variance and Standard Deviation of a Discrete Random Variable

6.2 The Binomial Probability Distribution Learning Objectives Determine whether a probability experiment is a binomial experiment Compute probabilities of binomial experiments Compute the mean and standard deviation of a binomial random variable Graph a binomial probability distribution

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.1 Determine Whether a Probability Experiment is a Binomial Experiment (1 of 5) Criteria for a Binomial Probability Experiment An experiment is said to be a binomial experiment if The experiment is performed a fixed number of times. Each repetition of the experiment is called a trial. The trials are independent. This means the outcome of one trial will not affect the outcome of the other trials. For each trial, there are two mutually exclusive (or disjoint) outcomes, success or failure. The probability of success is fixed for each trial of the experiment.

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.1 Determine Whether a Probability Experiment is a Binomial Experiment (2 of 5) 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.

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.1 Determine Whether a Probability Experiment is a Binomial Experiment (3 of 5) 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. Binomial experiment

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.1 Determine Whether a Probability Experiment is a Binomial Experiment (4 of 5) EXAMPLE Identifying Binomial Experiments Which of the following are binomial experiments? 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. Not a binomial experiment – not a fixed number of trials.

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.1 Determine Whether a Probability Experiment is a Binomial Experiment (5 of 5) EXAMPLE Identifying Binomial Experiments Which of the following are binomial experiment In a class of 30 students, 55% are female. The instructor randomly selects 4 students. The number X of females selected is recorded. Not a binomial experiment – the trials are not independent.

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.2 Compute Probabilities of Binomial Experiments (1 of 11) 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, 2008. 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.

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.2 Compute Probabilities of Binomial Experiments (2 of 11) EXAMPLE Constructing a Binomial Probability Distribution Compute probability of x = 0 (i.e., 0 successes).

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.2 Compute Probabilities of Binomial Experiments (3 of 11) EXAMPLE Constructing a Binomial Probability Distribution Compute probability of x = 1 (i.e., 1 success).

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.2 Compute Probabilities of Binomial Experiments (4 of 11) EXAMPLE Constructing a Binomial Probability Distribution Compute probability of x = 2 (i.e., 2 successes).

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.2 Compute Probabilities of Binomial Experiments (5 of 11) EXAMPLE Constructing a Binomial Probability Distribution Compute probability of x = 3 (i.e., 3 successes).

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.2 Compute Probabilities of Binomial Experiments (6 of 11) EXAMPLE Constructing a Binomial Probability Distribution Compute probability of x = 4 (i.e., 4 successes). x P(x) 0.00194 1 0.02926 2 0.16514 3 0.41415 4 0.3890

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.2 Compute Probabilities of Binomial Experiments (7 of 11) Binomial Probability Distribution Function The probability of obtaining x successes in n independent trials of a binomial experiment is given by

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.2 Compute Probabilities of Binomial Experiments (8 of 11) Phrase Math Symbol “at least” or “no less than” or “greater than or equal to” ≥ “more than” or “greater than” > “fewer than” or “less than” < “no more than” or “at most” or “less than or equal to ≤ “exactly” or “equals” or “is” =

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.2 Compute Probabilities of Binomial Experiments (9 of 11) 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?

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.2 Compute Probabilities of Binomial Experiments (10 of 11) 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 less than 4 have three or more cars?

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.2 Compute Probabilities of Binomial Experiments (11 of 11) 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 at least 4 have three or more cars?

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.3 Compute the Mean and Standard Deviation of a Binomial Random Variable (1 of 2) Mean (or Expected Value) and Standard Deviation of a Binomial Random Variable A binomial experiment with n independent trials and probability of success p has a mean and standard deviation given by the formulas

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.3 Compute the Mean and Standard Deviation of a Binomial Random Variable (2 of 2) 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.

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.4 Graph a Binomial Probability Distribution (1 of 15) EXAMPLE Graphing Binomial Probability Distributions Graph the binomial probability distribution with n = 8 and p = 0.15. Graph the binomial probability distribution with n = 8 and p = 0. 5. Graph the binomial probability distribution with n = 8 and p = 0.85.

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.4 Graph a Binomial Probability Distribution (2 of 15)

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.4 Graph a Binomial Probability Distribution (3 of 15)

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.4 Graph a Binomial Probability Distribution (4 of 15)

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.4 Graph a Binomial Probability Distribution (5 of 15) EXAMPLE Graphing Binomial Probability Distributions Graph the binomial probability distribution with n = 25 and p = 0.8. Comment on the shape of the distribution.

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.4 Graph a Binomial Probability Distribution (6 of 15)

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.4 Graph a Binomial Probability Distribution (7 of 15) EXAMPLE Graphing Binomial Probability Distributions Graph the binomial probability distribution with n = 50 and p = 0.8. Comment on the shape of the distribution.

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.4 Graph a Binomial Probability Distribution (8 of 15)

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.4 Graph a Binomial Probability Distribution (9 of 15) EXAMPLE Graphing Binomial Probability Distributions Graph the binomial probability distribution with n = 70 and p = 0.8. Comment on the shape of the distribution.

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.4 Graph a Binomial Probability Distribution (10 of 15)

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.4 Graph a Binomial Probability Distribution (11 of 15) For a fixed, p, as the number of trials n in a binomial experiment increases, the probability distribution of the random variable X becomes bell-shaped. As a rule of thumb, if np(1 − p) > 10, the probability distribution will be approximately bell-shaped.

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.4 Graph a Binomial Probability Distribution (12 of 15) Use the Empirical Rule to identify unusual observations in a binomial experiment. The Empirical Rule states that in a bell-shaped distribution about 95% of all observations lie within two standard deviations of the mean.

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.4 Graph a Binomial Probability Distribution (13 of 15) Use the Empirical Rule to identify unusual observations in a binomial experiment. 95% of the observations lie between μ − 2σ and μ + 2σ. Any observation that lies outside this interval may be considered unusual because the observation occurs less than 5% of the time.

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.4 Graph a Binomial Probability Distribution (14 of 15) 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?

6. 2 The Binomial Probability Distribution 6. 2 6.2 The Binomial Probability Distribution 6.2.4 Graph a Binomial Probability Distribution (15 of 15) EXAMPLE Using the Mean, Standard Deviation and Empirical Rule to Check for Unusual Results in a Binomial Experiment

6.3 The Poisson Probability Distribution Objectives Determine if a probability experiment follows a Poisson process Compute probabilities of a Poisson random variable Find the mean and standard deviation of a Poisson random variable

6. 3 The Poisson Probability Distribution 6. 3 6.3 The Poisson Probability Distribution 6.3.1 Determine If a Probability Experiment Follows a Poisson Process (1 of 2) A random variable X, the number of successes in a fixed interval, follows a Poisson process provided the following conditions are met. The probability of two or more successes in any sufficiently small subinterval is 0. The probability of success is the same for any two intervals of equal length. The number of successes in any interval is independent of the number of successes in any other interval provided the intervals are not overlapping.

6. 3 The Poisson Probability Distribution 6. 3 6.3 The Poisson Probability Distribution 6.3.1 Determine If a Probability Experiment Follows a Poisson Process (2 of 2) 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.

6. 3 The Poisson Probability Distribution 6. 3 6.3 The Poisson Probability Distribution 6.3.1 Compute Probabilities of a Poisson Random Variable (1 of 3) Poisson Probability Distribution Function If X is the number of successes in an interval of fixed length t, then the probability of obtaining x successes in the interval is

6. 3 The Poisson Probability Distribution 6. 3 6.3 The Poisson Probability Distribution 6.3.1 Compute Probabilities of a Poisson Random Variable (2 of 3) 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 (a) exactly three. Interpret the result. (b) fewer than three. Interpret the result. (c) at least three. Interpret the result.

6. 3 The Poisson Probability Distribution 6. 3 6.3 The Poisson Probability Distribution 6.3.1 Compute Probabilities of a Poisson Random Variable (3 of 3)

6. 3 The Poisson Probability Distribution 6. 3 6.3 The Poisson Probability Distribution 6.3.3 Find the Mean and Standard Deviation of a Poisson Random Variable (1 of 6) Mean and Standard Deviation of a Poisson Random Variable A random variable X that follows a Poisson process with parameter λ has mean (or expected value) and standard deviation given by the formulas

6. 3 The Poisson Probability Distribution 6. 3 6.3 The Poisson Probability Distribution 6.3.3 Find the Mean and Standard Deviation of a Poisson Random Variable (2 of 6) Poisson Probability Distribution Function If X is the number of successes in an interval of fixed length and X follows a Poisson process with mean μ, the probability distribution function for X is

6. 3 The Poisson Probability Distribution 6. 3 6.3 The Poisson Probability Distribution 6.3.3 Find the Mean and Standard Deviation of a Poisson Random Variable (3 of 6) 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?

6. 3 The Poisson Probability Distribution 6. 3 6.3 The Poisson Probability Distribution 6.3.3 Find the Mean and Standard Deviation of a Poisson Random Variable (4 of 6) EXAMPLE Mean and Standard Deviation of a Poisson Random Variable

6. 3 The Poisson Probability Distribution 6. 3 6.3 The Poisson Probability Distribution 6.3.3 Find the Mean and Standard Deviation of a Poisson Random Variable (5 of 6) EXAMPLE A Poisson Process? In 1910, Ernest Rutherford and Hans Geiger recorded the number of α-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 α-particles emitted? Number Detected Frequency 57 1 203 2 383 3 525 4 532 5 408 6 273 7 139 8 45 9 27 10 11

Expected Proportion if Poisson Expected Frequency if Poisson 6.3 The Poisson Probability Distribution 6.3.3 Find the Mean and Standard Deviation of a Poisson Random Variable (6 of 6) x Frequency, f x ∙ f Expected Proportion if Poisson Expected Frequency if Poisson 57 0.020866 54.4 1 203 0.080744 210.6 2 383 766 0.156225 407.4 3 525 1575 0.201511 525.5 4 532 2128 0.194943 508.4 5 408 2040 0.150872 393.5 6 273 1638 0.097303 253.8 7 139 973 0.053789 140.3 8 45 360 0.026018 67.9 9 27 243 0.011187 29.2 10 100 0.004329 11.3 11 6µ 66 0.001523 4.0 blank Sum: 2608 10092 µ = 3.869632