Sections 5.2 %26 5.3 Recitation Problems

Sections 5.2 %26 5.3 Recitation Problems - Solutions for...

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Solutions for Recitation Problems from Chapter 5: Problems 14 and 20 (also 19 from HW #6; 10 and 21) 14. Let X be the discrete r.v. of profit, in $K, with values x = -100, 0, 50, 100, 150, and 200. a. To be a valid probability distribution, prob (-100) + prob (0) + prob (50) + prob (100) + prob (150) + prob (200) = 1 So Prob ( x = 200) = 1 – [ prob ( x = -100) + prob ( x = 0) + prob ( x = 50) + prob ( x = 100) + prob ( x = 150)] = 1 – (0.10 + 0.20 + 0.30 + 0.25 + 0.10) = 1 – 0.95 = 0.05 The probability that MRA will have a $200,000 profit is 0.05. In other words, there is a 5% chance that MRA will have a $200,000 profit. The probability distribution for profit x is: x prob(x) -100 0.10 0 0.20 50 0.30 100 0.25 150 0.10 200 0.05 1.00 b. “Profit” means x > 0, so Prob (Profit) = prob ( x > 0) = prob ( x = 50) + prob ( x = 100) + prob ( x = 150) Chap 5 recitation problems 1 © Harvey Singer 2009
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+ prob ( x = 200) 200) = 0.30 + 0.25 + 0.10 + 0.05 = 0.70 The probability that MRA will have a profit is 0.70. In other words, there is a 70% chance that MRA will be profitable. c. The phrase “at least $100,000” means “$100,000 or more.” The discrete values of x that are “100,000 or more” are x = 100, 150, and 200. Prob (at least 100) = prob ( x ≥ 100) = prob ( x = 100) + prob ( x = 150) + prob ( x = 200) = 0.25 + 0.10 + 0.05 = 0.40 The probability that MRA will have a profit of at least $100,000 is 0.40. In other words, there is a 40% chance that MRA will have a profit of at least $100,000. (Another way of stating this: There is a 40% chance that MRA will have a profit of at not less than $100,000.) ********************************************************* New questions: 1. How much profit should be expected? Calculate expected value of x from the probability distribution: x prob(x) x*prob(x) -100 0.10 -10 0 0.20 0 50 0.30 15 100 0.25 25 150 0.10 15 200 0.05 10 1.00 55.00 Chap 5 recitation problems 2 © Harvey Singer 2009
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Expect a profit of $55,000. Of course, the profit may be more or it may be less, but as a first an best guess, expect a profit of $55K. Notes: a. The expected value is NOT the arithmetic average (mean) of the values of the r.v. x . That is, the expected value of x is NOT the 66.67, the arithmetic average of the values x = -100, 0, 50, 100, 150, and 200. This is because the values are not all equally likely, that is, they do not have the same chances of occurrence. For randomly sampled data, it is assumed that all the values are equally likely, as was done in Chapter 3 (although not stated at the time). But not here. These are not randomly sampled values. They are all possible values of the random variable X . Some of the values of x are more likely than others. As a result, the probability of a value, and not just the value itself, also drives what should be expected. b. The r.v.
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Sections 5.2 %26 5.3 Recitation Problems - Solutions for...

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