Discrete Probability Distributions Chapter 6 Discrete Probability Distributions

Discrete Random Variables Section 6.1 Discrete Random Variables

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. 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. 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. 3

(b) The number of leaves on a randomly selected oak tree. 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 4

The number of As earned in a section of statistics 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 As earned in a section of statistics (b) The time it takes to fly from California to Japan (c) The speed of the next car that passes a state trooper. Discrete; x = 0, 1, 2, …, 10 Continuous; t > 0 Continuous; t > 0 5

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

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 7

Rules for a Discrete Probability Distribution Let P(x) denote the probability that the random variable X equals x; then Σ P(x) = 1 0 ≤ P(x) ≤ 1 8

Is the following a probability distribution? 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 No. Σ P(x) = 0.97 9

Is the following a probability distribution? 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 No. P(x = 5) = –0.01 10

Is the following a probability distribution? 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 Yes. Σ P(x) = 1 and all probabilities are between 0 and 1. 11

EXAMPLE Drawing a Probability Histogram 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 movies streamed on Netflix each month. 12

EXAMPLE Drawing a Probability Histogram P(x) 0.01 1 0.10 2 0.38 3 0.58 Graph the discrete probability distribution given in the following table. 13

The Mean of a Discrete Random Variable The mean of a discrete random variable is given by the formula where x is the value of the random variable and P(x) is the probability of observing the value x. 14

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 15

Compute the mean of the probability distribution to the right. EXAMPLE Drawing a Probability Histogram x P(x) 0.01 1 0.10 2 0.38 3 0.58 Compute the mean of the probability distribution to the right. 16

Compute the mean of the probability distribution to the right. EXAMPLE Drawing a Probability Histogram x P(x) 1 0.20 2 0.25 3 0.10 4 0.14 5 0.31 Compute the mean of the probability distribution to the right. 17

The difference between and μX gets closer to 0 as n increases. 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 The difference between and μX gets closer to 0 as n increases. 18

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. 19

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As the number of trials of the experiment increases, the mean number of rentals approaches the mean of the probability distribution from previous example. 21

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. 22

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. x P(x) 530 0.99791 530 – 250,000 = -249,470 0.00209 Survives Does not survive E(X) = 530(0.99791) + (-249,470)(0.00209) = $7.50 The insurance expects to make $7.50 profit on every 49-year-old female it insures. 23

EXAMPLE Computing the Expected Value of a Discrete Random Variable A life insurance company sells a $250,000 1-year term life insurance policy to a 20-year-old male for $350. According to the National Vital Statistics Report the probability that the male survives the year is 0.998734. Compute and interpret expected value of this policy to the insurance company. x P(x) 350 0.998734 350 – 250,000 = -249,650 0.001266 Survives Does not survive E(X) = 350(0.998734) + (-249,650)(0.001266) = $33.50 The insurance company expects to earn an average profit of $33.50 on every 20-year-old male it insures for 1 year. 24

Standard Deviation of a Discrete Random Variable The standard deviation of a discrete random variable X is given by where x is the value of the random variable, μX is the mean of the random variable, and P(x) is the probability of observing a value of the random variable. 25

EXAMPLE. Computing the Variance and Standard Deviation 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 26

EXAMPLE. Computing the Variance and Standard EXAMPLE Computing the Variance and Standard Deviation of a Discrete Random Variable x μ P(x) 1.49 –1.49 2.2201 0.06 0.133206 1 –0.49 0.2401 0.58 0.139258 2 0.51 0.2601 0.22 0.057222 3 1.51 2.2801 0.1 0.22801 4 2.51 6.3001 0.03 0.189003 5 3.51 12.3201 0.01 0.123201 27

EXAMPLE. Computing the Variance and Standard Deviation EXAMPLE Computing the Variance and Standard Deviation of a Discrete Random Variable Compute the variance and standard deviation of the following probability distribution using technology. x P(x) 0.01 1 0.10 2 0.38 3 0.58 28

The Binomial Probability Distribution 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. 30

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. 31

Criteria for a Binomial Probability Experiment (ABRIDGED) An experiment is said to be a binomial experiment if Performed a fix number of times Trials are independent Only two possible outcomes Probability of success is constant 32

Which of the following are binomial experiments? EXAMPLE Identifying Binomial Experiments Which of the following are binomial experiments? (a) A player rolls a pair of fair die 10 times. The number X of 7’s rolled is recorded. Binomial experiment 33

Which of the following are binomial experiments? EXAMPLE Identifying Binomial Experiments Which of the following are binomial experiments? (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. Not a binomial experiment – not a fixed number of trials. 34

Which of the following are binomial experiments? EXAMPLE Identifying Binomial Experiments Which of the following are binomial experiments? (c) 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. 35

Which of the following are binomial experiments? EXAMPLE Identifying Binomial Experiments Which of the following are binomial experiments? (d) A probability experiment in which three cards are drawn from a deck without replacement and the number of aces is recorded. Not a binomial experiment – the trials are not independent. 36

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. 37

Compute probability of x = 0 (i.e., 0 successes). EXAMPLE Constructing a Binomial Probability Distribution Compute probability of x = 0 (i.e., 0 successes). 38

Compute probability of x = 1 (i.e., 1 success). EXAMPLE Constructing a Binomial Probability Distribution Compute probability of x = 1 (i.e., 1 success). 39

Compute probability of x = 2 (i.e., 2 successes). EXAMPLE Constructing a Binomial Probability Distribution Compute probability of x = 2 (i.e., 2 successes). 40

Compute probability of x = 3 (i.e., 3 successes). EXAMPLE Constructing a Binomial Probability Distribution Compute probability of x = 3 (i.e., 3 successes). 41

Compute probability of x = 4 (i.e., 4 successes). 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 42

EXAMPLE Constructing a Binomial Probability Distribution According to the American Red Cross, 7% of people in the United States have blood type O-negative. A simple random sample of size 3 is obtained, and the number of people X with blood type O-negative is recorded. Construct a probability distribution for the random variable X using a tree diagram. 43

Binomial Probability Distribution Function The probability of obtaining x successes in n independent trials of a binomial experiment is given by where p is the probability of success. 44

“at least” or “no less than” or “greater than or equal to” ≥ 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” = 45

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? 46

EXAMPLE Using the Binomial Probability Distribution Function According to the Experian Automotive, 35% of all car-owning households 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? 47

EXAMPLE Using the Binomial Probability Distribution Function According to the Experian Automotive, 35% of all car-owning households 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? 48

Exactly 5 are wireless-only? Fewer than 3 are wireless-only? EXAMPLE Using the Binomial Probability Distribution Function According to CTIA, 41% of all U.S. households are wireless only households. In a random sample of 20 households, what is the probability that: Exactly 5 are wireless-only? Fewer than 3 are wireless-only? At least 3 are wireless-only? The number of households that are wireless-only is between 5 and 7, inclusive? 49

exactly 10 favor the death penalty? EXAMPLE Using Technology According to the Gallup Organization, 65% of adult Americans are in favor of the death penalty for individuals convicted of murder. In a random sample of 15 adult Americans, what is the probability that (using technology): exactly 10 favor the death penalty? no more than 6 favor the death penalty? Binompdf(15,.65,10) Binomcdf(15,.65,6) 50

exactly 15 feel the state of morals is poor? EXAMPLE Using Technology In a poll, 45% of adult Americans believe that the overall state of moral values in the United States is poor. Suppose a survey of a random sample of 25 adult Americans is conducted on their views of moral values in America. What is the probability that (using technology): exactly 15 feel the state of morals is poor? no more than 10 feel the state of morals is poor? more than 16 feel the state of morals is poor? 13 or 14 believe the state of morals is poor? Would it be unusual to find 20 or more who believe the state of morals is poor? 51

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 52

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. 53

EXAMPLE Finding the Mean and Standard Deviation of a Binomial Random Variable According to CTIA, 41% of all U.S. households are wireless-only households. In a simple random sample of 300 households, determine the mean and standard deviation number of wireless-only households. 54

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. 55

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. 56

95% of the observations lie between μ – 2σ and μ + 2σ. 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. 57

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 ? 58

The result is unusual since 162 > 159.1 EXAMPLE Using the Mean, Standard Deviation and Empirical Rule to Check for Unusual Results in a Binomial Experiment Check np(1-p) = (400)(.35)(.65) = 91 >10 The result is unusual since 162 > 159.1 59

EXAMPLE Using the Mean, Standard Deviation and Empirical Rule to Check for Unusual Results in a Binomial Experiment According to CTIA, 41% of all U.S. households are wireless-only. In a simple random sample of 300 households, 143 were wireless-only. Is this result unusual? 60