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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 16- 1

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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 16 Random Variables

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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 16- 3 Expected Value: Center A random variable assumes a value based on the outcome of a random event. We use a capital letter, like X, to denote a random variable. A particular value of a random variable will be denoted with a lower case letter, in this case x.

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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 16- 4 Expected Value: Center (cont.) There are two types of random variables: Discrete random variables can take one of a finite number of distinct outcomes. Example: Number of siblings Continuous random variables can take any numeric value within a range of values. Example: Weight of books (in lbs.) this term

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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 16- 5 Consider an experiment whose population is the set of all Kennesaw State University students. Different random variables are number of classes the student is enrolled in this semester; Height in inches; annual income in dollars; GPA; number of miles driven to campus. Which of these variables are discrete and which are continuous?

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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 16- 6 Expected Value: Center (cont.) A probability model for a random variable consists of: The collection of all possible values of a random variable, and the probabilities that the values occur. Of particular interest is the value we expect a random variable to take on, notated μ (for population mean) or E(X) for expected value.

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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 16- 7 Expected Value: Center (cont.) The expected value of a (discrete) random variable can be found by summing the products of each possible value by the probability that it occurs: Note: Be sure that every possible outcome is included in the sum and verify that you have a valid probability model to start with.

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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 16- 8 Expected Value: Center (cont.) A box contains four slips of paper containing the numbers -2, -1, 0, 2. A game consists of randomly selecting a slip of paper from the box and receiving that amount in dollars. You may play this game as many times as you wish. Will you play this game? Compute the expected value. If you play this game 100 times, what will the monetary result be?

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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 16- 9 Expected Value: Center (cont.) A box contains four slips of paper containing the numbers -10, 1, 2, 2. Will you play this game? Compute the expected value.

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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 16- 10 Expected Value: Center (cont.) A box contains fourteen million slips of paper. One slip contains the value five million. All the other slips contain a -1. Will you play this game?

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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 16- 11 Expected Value: Center (cont.) You pay $1 to play a game. The game consists of rolling a pair of dice. If you observe a sum of 7 or 11 you receive $4. If not, you receive nothing. How is receiving $4 different from winning $4? What is the expected value of this game?

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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 16- 12 First Center, Now Spread… For data, we calculated the standard deviation by first computing the deviation from the mean and squaring it. We do that with random variables as well. The variance for a random variable is: The standard deviation for a random variable is:

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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 16- 13 First Center, Now Spread… You draw a card from a deck. If you get a club you get $5. If you get an Ace you get $10. Create a probability model for this game. Compute expected value and standard deviation for this game.

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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 16- 14 You draw a card from a deck. If you get a club you get nothing. If you get red card you get $10 If you get a spade you get $15 and get to select another card (without replacement). If the second is another spade you receive an additional $20. You receive nothing for any non-spade card. Create a probability model for this game. Compute expected value and standard deviation for this game.

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