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STA 301 - Ch 04 - pp 55-63

STA 301 - Ch 04 - pp 55-63 - Chapter 4 Mathematical...

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Chapter 4: Mathematical Expectation 55 CHAPTER 4 — MATHEMATICAL EXPECTATION In Chapter 1 we examined numerical summaries for distributions of data sets. In Chapter 4, we extend these ideas to random variables. MEAN OF A DISCRETE RANDOM VARIABLE Example : Suppose a driver’s education instructor recorded the number of attempts each of his 50 students required to pass the road test. The data are summarized below: Number of Attempts(x) Frequency Relative Frequency 1 30 30 / 50 = .60 2 10 10 / 50 = .20 3 5 5 / 50 = .10 4 3 3 / 50 = .06 5 2 2 / 50 = .04 Total 50 50 / 50 = 1.00 Two methods to calculate the mean of the data set: Using frequencies Using relative frequencies Now, suppose a student is randomly selected from this class. Let X be the random variable representing the number of attempts it took the student to pass the road test. Then the probability function is: ( 29 = = = = = = 5 04 . 4 06 . 3 10 . 2 20 . 1 60 . x if x if x if x if x if x f Calculating the mean of a discrete random variable is similar to calculating the mean of a data set using relative frequencies.
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Chapter 4: Mathematical Expectation 56 The mean or expected value of a discrete random variable X is given by: ( ( = = x x x xf X E μ Calculate the expected value of X in the above example: Example : (p. 109, Ex. 4.2) In a gambling game a man is paid $5 if he gets all heads or all tails when three coins are tossed, and he will pay out $3 if either one or two heads show. What is his expected gain?
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