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chapter 2-Discrete Probability Distributions

# chapter 2-Discrete Probability Distributions - CHAPTER 2...

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CHAPTER 2 DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS 1. Random Variable 2. Probability Distribution 3. Expected Value of a Random Variable 4. The Variance And Standard Deviation of a Random Variable 4.1. Arithmetic Properties Of Expected Value And Variance 4.1.1. Standardized Random Variable 5. The Binomial Distribution 5.1. Cumulative Probability Distribution 5.2. How to Use Excel to Find Binomial Probabilities 5.3. Expected Value and Variance of the Binomial Distribution 1. Random Variable A random variable is a variable whose values are determined through a random experiment or process. In other words, a random variable is a variable whose value cannot be predicted exactly. The value is not known in advance; it is not known until after the random experiment is conducted. Example 1 As a random experiment, toss a coin. This random experiment has two outcomes: H and T. Let’s assign the value 0 to H, (H = 0), and 1 to T, (T = 1). Counting the number of 1’s in each outcome provides the values assigned to x . If you toss two coins, then the number of tails is either 0, 1, or 2: Outcomes of the random experiment Values of random variable x Number of tails (0,0) 0 (0,1), (1,0) 1 (1,1) 2 Example 2 When you toss a pair of dice, let x denote the sum of the number of dots appearing on top. These numbers, as they appear in the top row below, are assigned to x through the outcomes of the random experiment. The outcomes are shown below. Outcomes of the random experiment Values of random variable x Sum of dots (1,1) 2 (1,2), (2,1) 3 (1,3), (2, 2), (3,1) 4 (1,4), (2,3), (3,2), (4,1) 5 (1,5), (2,4), (3,3), (4,2), (5,1) 6 (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) 7 (2,6), (3,5), (4,4), (5,3), (6,2) 8 (3,6), (4,5), (5,4), (6,3) 9 (4,6), (5,5), (6,4) 10 (5,6), (6,5) 11 Chapter 2—Discrete Random Variables and Probability Distributions Page 1 of 19

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(6,6) 12 Example 3 When you are guessing the answers to a set of 5 multiple-choice questions, you are conducting a random experiment. Assign 0 to the incorrect answer and 1 to the correct answer for each question, and let x denote the number of possible correct answers: 0, 1, 2, 3, 4, 5. These numbers are assigned to x through the outcomes of the random experiment as shown below. There are 32 possible outcomes in this random experiment. 1 Outcomes of the random experiment x Correct guesses (0,0,0,0,0) 0 (1,0,0,0,0), (0,1,0,0,0), (0,0,1,0,0), (0,0,0,1,0), (0,0,0,0,1), 1 (1,1,0,0,0), (1,0,1,0,0), (1,0,0,1,0), (1,0,0,0,1), (0,1,1,0,0), (0,1,0,1,0), (0,1,0,0,1), (0,0,1,1,0), (0,0,1,0,1), (0,0,0,1,1) 2 (1,1,1,0,0), (1,1,0,1,0), (1,1,0,0,1), (1,0,1,1,0), (1,0,1,0,1), (1,0,0,1,1), (0,1,1,1,0), (0,1,1,0,1), (0,1,0,1,1), (0,0,1,1,1) 3 (1,1,1,1,0), (1,1,1,0,1), (1,1,0,1,1), (1,0,1,1,0), (0,1,1,1,1) 4 (1,1,1,1,1) 5 2. Probability Distribution The probability distribution of a random variable is the listing of all possible values of the random variable along with the probability corresponding to each value. Since a probability distribution lists all possible values of the random variable, then the sum of the probabilities must equal one (1). Example 4 Let x be the random variable denoting the number of tails when tossing two coins as in Example 1. Write the probability distribution of x . As Example 1 showed, there are four outcomes when tossing two coins, each assigning a discrete value to x . Since each outcome is equally likely then the probability distribution is: x f ( x ) 0 0.25 1 0.50 2 0.25 1.00 The graph of the above probability distribution is: 1 Each trial has two outcomes (failure or success—0 or 1). The experiment has five trials.
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chapter 2-Discrete Probability Distributions - CHAPTER 2...

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