ESSENTIAL STATISTICS 2 E William Navidi and Barry
ESSENTIAL STATISTICS 2 E William Navidi and Barry Monk ©Mc. Graw-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 Mc. Graw-Hill Education.
Random Variables Section 5. 1 ©Mc. Graw-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 ©Mc. Graw-Hill Education.
Objective 1 Distinguish between discrete and continuous random variables ©Mc. Graw-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. ©Mc. Graw-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. ©Mc. Graw-Hill Education.
Objective 2 Determine a probability distribution for a discrete random variable ©Mc. Graw-Hill Education.
Probability Distribution • ©Mc. Graw-Hill Education.
Example 1: Probability Distribution Decide if the following represents a probability distribution. • ©Mc. Graw-Hill Education. x 1 2 3 P(x) 0. 25 0. 65 – 0. 30 4 0. 11
Example 2: Probability Distribution Decide if the following represents a probability distribution. x – 1 – 0. 5 0 P(x) 0. 17 0. 25 0. 31 0. 5 1 0. 22 0. 05 This is a probability distribution. All the probabilities are between 0 and 1, and they add up to 1. ©Mc. Graw-Hill Education.
Example 3: Probability Distribution Decide if the following represents a probability distribution. • ©Mc. Graw-Hill Education. x 1 10 100 P(x) 1. 02 0. 31 0. 90 1000 0. 43
Example: Computing Probabilities (Part a) • • ©Mc. Graw-Hill Education. x 0 1 2 P(x) 0. 23 0. 41 0. 27 3 4 0. 08 0. 01
Example: Computing Probabilities (Part b) • • ©Mc. Graw-Hill Education. x 0 1 2 P(x) 0. 23 0. 41 0. 27 3 4 0. 08 0. 01
Example: Computing Probabilities (Part c) • • ©Mc. Graw-Hill Education. x 0 1 2 P(x) 0. 23 0. 41 0. 27 3 4 0. 08 0. 01
Objective 3 Describe the connection between probability distributions and populations ©Mc. Graw-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. ©Mc. Graw-Hill Education.
Example: Connection with Populations • • ©Mc. Graw-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. • ©Mc. Graw-Hill Education. x 1. 50 2. 00 4. 00 P(x) 0. 423 0. 142 0. 057 4. 50 0. 378
Objective 4 Construct a probability histogram for a discrete random variable ©Mc. Graw-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. ©Mc. Graw-Hill Education. x P(x) 0 0. 03125 1 0. 15625 2 0. 31250 3 0. 31250 4 0. 15625 5 0. 03125
Objective 5 Compute the mean of a discrete random variable ©Mc. Graw-Hill Education.
Mean of a Random Variable • ©Mc. Graw-Hill Education.
Example: Mean of a Random Variable • • ©Mc. Graw-Hill Education. x P(x) 0 0. 2 1 0. 5 2 0. 2 3 0. 1
Expected Value • ©Mc. Graw-Hill Education.
Example: Expected Value • Solution: The probability distribution is as follows. Note that 30 is negative since it represents a loss. • ©Mc. Graw-Hill Education. x P(x) – 30 0. 4 20 0. 5 40 0. 1
Objective 6 Compute the variance and standard deviation of a discrete random variable ©Mc. Graw-Hill Education.
Variance/Standard Deviation of a Random Variable • ©Mc. Graw-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 L 1 and the associated probabilities in L 2. Step 2: Press STAT and highlight the CALC menu and select 1 -Var Stats with L 1 and L 2 as the arguments. ©Mc. Graw-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. • ©Mc. Graw-Hill Education. x P(x) 0 0. 2 1 0. 5 2 0. 2 3 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 ©Mc. Graw-Hill Education.
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