ELEMENTARY STATISTICS 3E William Navidi and Barry Monk McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education. Random Variables Section 6.1 McGraw-Hill Education. Objectives 1. Distinguish between discrete and continuous random variables 2. Determine a probability distribution for a discrete random

variable 3. Describe the connection between probability distributions and populations 4. Construct a probability histogram for a discrete random variable 5. Compute the mean of a discrete random variable 6. Compute the variance and standard deviation of a discrete random variable McGraw-Hill Education. Objective 1 Distinguish between discrete and continuous random variables

McGraw-Hill Education. Random Variable If we roll a fair die, the possible outcomes are the numbers 1, 2, 3, 4, 5, and 6, and each of these numbers has probability 1/6. Rolling a die is a probability experiment whose outcomes are numbers. The outcome of such an experiment is called a random variable. A random variable is a numerical outcome of a probability experiment. McGraw-Hill Education.

Discrete and Continuous Random Variables Discrete random variables are random variables whose possible values can be listed. Examples include: The number that comes up on the roll of a die. The number of siblings a randomly chosen person has. Continuous random variables are random variables that can take on any value in an interval. Examples include: The height of a randomly chosen college student. The amount of electricity used to light a randomly chosen classroom. McGraw-Hill Education. Objective 2

Determine a probability distribution for a discrete random variable McGraw-Hill Education. Probability Distribution A probability distribution for a discrete random variable specifies the probability for each possible value of the random variable. Properties: for every possible McGraw-Hill Education.

Example 1: Probability Distribution Decide if the following represents a probability distribution. x 1 2 3 4 P(x) 0.25 0.65 0.30 0.11

This is not a probability distribution. (3) is not between 0 and 1. McGraw-Hill Education. Example 2: Probability Distribution Decide if the following represents a probability distribution. x 1 0.5 0 0.5 1

P(x) 0.17 0.25 0.31 0.22 0.05 This is a probability distribution. All the probabilities are between 0 and 1, and they add up to 1. McGraw-Hill Education. Example 3: Probability Distribution Decide if the following represents a probability distribution.

x 1 10 100 1000 P(x) 1.02 0.31 0.90 0.43 This is not a probability distribution. (1) is not between 0 and 1.

McGraw-Hill Education. Example: Computing Probabilities (Part a) Four patients have made appointments to have their blood pressure checked at a clinic. Let be the number of them that have high blood pressure. The probability distribution of is as follows. x 0 1 2 3

4 P(x) 0.23 0.41 0.27 0.08 0.01 a) Find (2 or 3) Solution: The events 2 and 3 are mutually exclusive, since they cannot both happen. We use the Addition Rule for Mutually Exclusive events:

(2 or 3) = (2) + (3) = 0.27 + 0.08 = 0.35 McGraw-Hill Education. Example: Computing Probabilities (Part b) Four patients have made appointments to have their blood pressure checked at a clinic. Let be the number of them that have high blood pressure. The probability distribution of is as follows. x 0 1 2

3 4 P(x) 0.23 0.41 0.27 0.08 0.01 b) Find (More than 1) Solution: More than 1 means 2 or 3 or 4. We use the Addition Rule for Mutually Exclusive events:

(More than 1) = (2 or 3 or 4) = 0.27 + 0.08 + 0.01 = 0.36 McGraw-Hill Education. Example: Computing Probabilities (Part c) Four patients have made appointments to have their blood pressure checked at a clinic. Let be the number of them that have high blood pressure. The probability distribution of is as follows. x 0 1

2 3 4 P(x) 0.23 0.41 0.27 0.08 0.01 c) Find (At least 1) Solution: We use the Rule of Complements. Recall that the complement of At

least one is none: (At least one) = 1 (0) = 1 0.23 = 0.77 McGraw-Hill Education. Objective 3 Describe the connection between probability distributions and populations McGraw-Hill Education. Probability Distributions and Populations Statisticians are interested in studying samples drawn from populations. Random variables are important because when

an item is drawn from a population, the value observed is the value of a random variable. The probability distribution of the random variable tells how frequently we can expect each of the possible values of the random variable to turn up in the sample. McGraw-Hill Education. Example: Connection with Populations An airport parking facility contains 1000 parking spaces. Of these, 142 are covered long-term spaces that cost $2.00 per hour, 378 are covered short-term spaces that cost $4.50 per hour, 423 are uncovered long-term spaces that cost $1.50 per hour, and 57 are uncovered short-term spaces that cost $4.00 per hour. A parking

space is selected at random. Let represent the hourly parking fee for the randomly sampled space. Find the probability distribution of . Solution: To find the probability distribution, we must list the possible values of and then find the probability of each of them. The possible values of are 1.50, 2.00, 4.00, 4.50. Next, we find their probabilities. McGraw-Hill Education. Example: Connection with Populations (Continued) An airport parking facility contains 1000 parking spaces. Of these, 142 are covered long-term spaces that cost $2.00 per hour, 378 are covered shortterm spaces that cost $4.50 per hour, 423 are uncovered long-term spaces that cost $1.50 per hour, and 57 are uncovered short-term spaces that cost

$4.00 per hour. x 1.50 2.00 4.00 4.50 McGraw-Hill Education. P(x) 0.423 0.142

0.057 0.378 is a Random Variable Often when we draw a sample, we compute the sample mean . The quantity is a random variable, because its value is different for different samples. The probability distribution for is usually difficult to compute. We will learn a way to approximate the probability distribution of the sample mean when the sample size is large in a later chapter. McGraw-Hill Education.

Objective 4 Construct a probability histogram for a discrete random variable McGraw-Hill Education. Probability Histograms Probability distributions can be represented with histograms to visualize the distribution. Example: The following presents the probability distribution and histogram for the number of boys in a family of five children, using the assumption that boys and girls are equally likely and that births are independent

events. x P(x) 0 1 2 3 4 5 McGraw-Hill Education. 0.03125 0.15625 0.31250

0.31250 0.15625 0.03125 Objective 5 Compute the mean of a discrete random variable McGraw-Hill Education. Mean of a Random Variable Recall that the mean is a measure of center. The mean of a random variable provides a measure of center for the probability distribution of a random variable.

To find the mean of a discrete random variable, multiply each possible value by its probability, then add the products: McGraw-Hill Education. Example: Mean of a Random Variable A computer monitor is composed of a very large number of points of light called pixels. It is not uncommon for a few of these pixels to be defective. Let represent the number of defective pixels on a randomly chosen monitor. The probability distribution of is as follows. Find the mean number of defective pixels. The

mean is If we imagine each rectangle in the probability histogram to be a weight, the mean is the point at which the histogram would balance. McGraw-Hill Education. x 0 1 2 3

P(x) 0.2 0.5 0.2 0.1 Expected Value There are many occasions on which people want to predict how much they are likely to gain or lose if they make a certain decision or take a certain action. Often, this is done by computing the mean of a random variable. In such situations, the mean is sometimes called the expected value and is denoted by . If the expected value

is positive, it is an expected gain, and if it is negative, it is an expected loss. McGraw-Hill Education. Example: Expected Value mineral economist estimated that a particular venture had probability A 0.4 of a $30 million loss, probability 0.5 of a $20 million profit, and probability 0.1 of a $40 million profit. Let represent the profit. Find the probability distribution of the profit and the expected value of the profit. Does this venture represent an expected gain or an expected loss? Solution: The probability distribution is as follows.

Note that 30 is negative since it represents a loss. expected value is The There is an expected gain of $2 million. McGraw-Hill Education. x 30 20 40 P(x)

0.4 0.5 0.1 Objective 6 Compute the variance and standard deviation of a discrete random variable McGraw-Hill Education. Variance/Standard Deviation of a Random Variable The variance and standard deviation provide a measure of spread for the probability distribution of a random variable.

The variance of a discrete random variable is given by or, equivalently (and easier to compute by hand) The standard deviation of a discrete random variable is the square root of the variance: McGraw-Hill Education. Mean/Standard Deviation on the TI-84 PLUS The mean and standard deviation of a random variable can be found on the TI-84 PLUS calculator with the following steps: Step 1: Enter the values of the random variable into L1 and the associated probabilities in L2.

Step 2: Press STAT and highlight the CALC menu and select 1-Var Stats with L1 and L2 as the arguments. McGraw-Hill Education. Example: Mean/Standard Dev. on the TI-84 Compute the mean and standard deviation using the TI-84 PLUS. Solution: We first enter values of the random variable and the associated probabilities into the data editor.

We run the 1-Var Stats L1, L2 command to find that and . McGraw-Hill Education. x 0 1 2 3 P(x) 0.2 0.5

0.2 0.1 You Should Know . . . The difference between discrete and continuous random variables How to determine the probability distribution for a discrete random variable How to construct a probability distribution for a population How to construct a probability histogram How to compute the mean, variance, and standard deviation of a discrete random variable

McGraw-Hill Education. The Binomial Distribution Section 6.2 McGraw-Hill Education. Objectives 1. Determine whether a random variable is binomial 2. Determine the probability distribution of a binomial random variable 3. Compute binomial probabilities 4. Compute the mean and variance of a binomial random variable

McGraw-Hill Education. Objective 1 Determine whether a random variable is binomial McGraw-Hill Education. Binomial Distribution Suppose that your favorite fast food chain is giving away a coupon with every purchase of a meal. Twenty percent of the coupons entitle you to a free hamburger, and the rest of them say better luck next time. Ten of you order lunch at this restaurant. Suppose we want to know the probability that three of you win a free

hamburger? In general, if we let be the number of people out of ten that win a free hamburger. What is the probability distribution of ? In this section, we will learn that has a distribution called the binomial distribution, which is one of the most useful probability distributions. McGraw-Hill Education. Conditions for a Binomial Distribution In the problem just described, each time we examine a coupon, we call it a trial, so there are 10 trials. When a coupon is good for a free hamburger, we will call it a success. The random variable represents the number of successes in 10 trials. A random variable that represents the number of successes in a series of trials

has a probability distribution called the binomial distribution. The conditions are: A fixed number of trials are conducted. There are two possible outcomes for each trial. One is labeled success and the other is labeled failure. The probability of success is the same on each trial. The trials are independent. This means that the outcome of one trial does not affect the outcomes of the other trials. The random variable represents the number of successes that occur. Notation: = number of trials, = probability of a success McGraw-Hill Education. Example 1: Binomial Experiment A fair coin is tossed ten times. Let be the number of times

the coin lands heads. Is this a binomial experiment? This is a binomial experiment. Each toss of the coin is a trial. There are two possible outcomes, heads and tails. Since represents the number of heads, heads counts as a success. The trials are independent, because the outcome of one coin toss does not affect the other tosses. McGraw-Hill Education. Example 2: Binomial Experiment Five basketball players each attempt a free throw. Let be the number of free throws made. Is this a binomial

experiment? This is not a binomial experiment. The probability of success (making a shot) differs from player to player, because they will not all be equally skilled at making free throws. McGraw-Hill Education. Example 3: Binomial Experiment Ten cards are in a box. Five are red and five are green. Three of the cards are drawn at random. Let be the number of red cards drawn. Is this a binomial experiment? This is not a binomial experiment because the trials are not independent.

McGraw-Hill Education. Objective 2 Determine the probability distribution of a binomial random variable McGraw-Hill Education. The Binomial Probability Distribution Consider the binomial experiment of tossing 3 times a biased coin that has probability 0.6 of coming up heads. Let be the number of heads that come up. If we want to compute (2), the probability that exactly 2 of the tosses are heads, there are 3 arrangements of two heads in three tosses: HHT,

HTH, THH. The probability of HHT is (HHT) = (0.6)(0.6)(0.4) = (0.6)2(0.4). Similarly, we find that (HTH) = (THH) = (0.6)2(0.4). Now, (2) = (HHT or HTH or THH) = 3(0.6)2(0.4), by the Addition Rule. Examining this result, we see the number 3 represents the number of arrangements of two successes (heads) and one failure (tails). In general, this number will be the number of arrangements of successes in trials, which is . The number 0.6 is the success probability which has an exponent of 2, the number of successes . The number 0.4 is the failure probability which has an exponent of 1, which is the number of failures, . McGraw-Hill Education. The Binomial Probability Distribution Formula In general, for a binomial random variable ,

The possible values of the random variable are 0, 1, , . McGraw-Hill Education. Objective 3* Compute binomial probabilities *(Hand Computation) McGraw-Hill Education. Example: Binomial (Hand Computation) The Pew Research Center recently reported that approximately 30% of Internet users in the United States use the image sharing website Pinterest. Suppose a simple random sample of 15 Internet users is taken.

Use the binomial probability distribution to find the following probabilities. a) Find the probability that exactly four of the sampled people use Pinterest. b) Find the probability that fewer than three of the people use Pinterest. c) Find the probability that more than one person uses Pinterest. d) Find the probability that the number of people who use Pinterest is between 1 and 4, inclusive. McGraw-Hill Education. Example: Part a) (Hand Comp.) Note that = 15 and = 0.3. a) Find the probability that exactly four of the sampled people use

Pinterest. We use the binomial probability distribution with = 4: McGraw-Hill Education. Example: Part b) (Hand Comp.) b) Find the probability that fewer than three of the people use Pinterest. The possible numbers of people that are fewer than three are 0, 1, and 2: P (0 or 1 or 2) = 0.3)15-1 15

C0(0.3)0 (1 0.3)15-0 + 15C1(0.3)1 (1 + 15C2(0.3)2 (1 0.3)15-2 = 0.0047+0.0305 + 0.0916 = 0.127 McGraw-Hill Education. Example: Part c) (Hand Comp.) c) Find the probability that more than one person uses Pinterest. We use the Rule of Complements. The complement of more than 1 is 1 or fewer or equivalently, 0 or 1.

The probability of the 0 or 1 is: P (0 or 1) = 15C0(0.3)0 (1 0.3)15-0 + 15C1(0.3)1 (1 0.3)15-1 = 0.0047+0.0305 = 0.035 Now, use the Rule of Complements: McGraw-Hill Education. Example: Part d) (Hand Comp.) d) Find the probability that the number of people who use Pinterest is between 1 and 4, inclusive. Between 1 and 4 inclusive means, 1, 2, 3, or 4.

P (1 or 2 or 3 or 4) = 0.3)15-2 C1(0.3)1 (10.3)15-1 + 15 + C2(0.3)2 (1 15

C3(0.3)3 (10.3)15-3 + 15 C4(0.3)4 15 (10.3)15-4 = 0.0305 + 0.0916 + 0.1700 + 0.2186 = 0.511 McGraw-Hill Education.

Objective 3** Compute binomial probabilities **(Tables) McGraw-Hill Education. Example: Binomial (Tables) The Pew Research Center recently reported that approximately 30% of Internet users in the United States use the image sharing website Pinterest. Suppose a simple random sample of 15 Internet users is taken. Use the binomial probability distribution to find the following probabilities. a) Find the probability that exactly four of the sampled people use

Pinterest. b) Find the probability that fewer than three of the people use Pinterest. c) Find the probability that more than one person uses Pinterest. d) Find the probability that the number of people who use Pinterest is between 1 and 4, inclusive. McGraw-Hill Education. Example: Part a) (Tables) Note that = 15 and = 0.3. a) Find the probability that exactly four of the sampled people use Pinterest.

McGraw-Hill Education. = 0.219 Example: Part b) (Tables) b) Find the probability that fewer than three of the people use Pinterest. McGraw-Hill Education. Example: Part c) (Tables) c) Find the probability that more than one person uses Pinterest.

We use the Rule of Complements. The complement of more than 1 is 1 or fewer or equivalently, 0 or 1. The probability of the 0 or 1 is: P (0 or 1) = 0.005+0.031 = 0.036 Now, use the Rule of Complements: McGraw-Hill Education. Example: Part d) (Tables) d) Find the probability that the number of people who use Pinterest is between 1 and 4, inclusive.

McGraw-Hill Education. Objective 3*** Compute binomial probabilities ***(TI-84 PLUS) McGraw-Hill Education. Binomial Probabilities on the TI-84 PLUS In the TI-84 PLUS Calculator, there are two primary commands for computing binomial probabilities. These are binompdf and binomcdf. These commands are on the DISTR (distributions) menu accessed by pressing 2nd, VARS.

The binompdf command is used when finding the probability that the binomial random variable is equal to a specific value, . The binomcdf command is used when finding the probability that the binomial random variable is less than or equal to a specified value, . McGraw-Hill Education. binompdf and binomcdf Commands binompdf To compute the probability that the random variable equals the

value given the parameters and , use the binompdf command with the following format: binompdf(n,p,x) binomcdf To compute the probability that the random variable is less than or equal to the value given the parameters and , use the binomcdf command with the following format: binomcdf(n,p,x) McGraw-Hill Education. Example: Binomial (TI-84 PLUS) The Pew Research Center recently reported that approximately 30% of

Internet users in the United States use the image sharing website Pinterest. Suppose a simple random sample of 15 Internet users is taken. Use the binomial probability distribution to find the following probabilities. a) Find the probability that exactly four of the sampled people use Pinterest. b) Find the probability that fewer than three of the people use Pinterest. c) Find the probability that more than one person uses Pinterest. d) Find the probability that the number of people who use Pinterest is between 1 and 4, inclusive. McGraw-Hill Education. Example: Part a) (TI-84 PLUS)

Note that = 15 and = 0.3. a) Find the probability that exactly four of the sampled people use Pinterest. Since we are finding the probability that equals 4, we use the binompdf command with = 15, = 0.3, and = 4. We find the probability that exactly four people use Pinterest is 0.219. McGraw-Hill Education. Example: Part b) (TI-84 PLUS) b) Find the probability that fewer than three of the people use Pinterest.

The binomcdf command computes the probability that there are less than or equal to successes. The event fewer than three is equivalent to less than or equal to two. We run the command binomcdf(15, 0.3, 2) to find that the probability that fewer than three people who use Pinterest is 0.1268. McGraw-Hill Education. Example: Part c) (TI-84 PLUS) c) Find the probability that more than one person uses Pinterest. We use the Rule of Complements. The complement of more than 1 is 1 or fewer. We use the command binomcdf(15, 0.3, 1) to first

find the probability that 1 or fewer people use Pinterest, and then subtract this value from 1 to find the probability that more than one person uses Pinterest. The result is approximately 0.965. McGraw-Hill Education. Example: Part d) (TI-84 PLUS) d) Find the probability that the number of people who use Pinterest is between 1 and 4, inclusive. Because (Between 1 and 4) = (4 or less) (0), we can find the probability using the commands binomcdf(15, 0.30, 4) binompdf(15, 0.30, 0) The result is approximately 0.511.

McGraw-Hill Education. Objective 4 Compute the mean and variance of a binomial random variable McGraw-Hill Education. Mean, Variance, and Standard Deviation Let be a binomial random variable with trials and success probability . Then the mean of is

The variance of is The standard deviation of is McGraw-Hill Education. Example: Mean and Standard Deviation The probability that a new car of a certain model will require repairs during the warranty period is 0.15. A particular dealership sells 80 such cars. Let be the number that will require repairs during the warranty period. Find the mean and standard deviation of . Solution: There are = 80 trials, with success probability = 0.15. The mean is .

The standard deviation is The interpretation of the mean is that in the long run, 12 out of every 100 cars will require repairs. The standard deviations measures the spread in the distribution of the number of cars that will require repairs. McGraw-Hill Education. You Should Know . . . How to determine whether a random variable is binomial The notation for a binomial experiment How to determine the probability distribution of a binomial random variable How to compute binomial probabilities How to compute the mean and variance of a binomial

random variable McGraw-Hill Education.