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Unformatted text preview: Chapter 4 Mathematical Expectation 4.1 Expected value of a discrete random variable Examples: (a) Given 1 , 2 , 3 , 4 , 5 , 6. What is the average? (b) Toss an unbalanced die 100 times. Lands on 1 2 3 4 5 6 Probab. .1 .3 .1 .2 .2 .1 Suppose we shall win $ i if the die lands on i . What is our “ideal” average winnings per toss? Solution : Definition 4.1 If X is a discrete r.v. taking on values x 1 , x 2 , ... and f ( x ) is the probability density function of X , the expected value ( or mathematical expectation or mean ) of X , denoted by E ( X ) , or μ X , is defined by E ( X ) = X i x i f ( x i ) = X x xf ( x ) 41 • The expected value of X is a weighted average of the possible values that X can take on, each value being weighted by the probability that X assumes. Example 4.1 The probability density function of X is given by x 1 f ( x ) 1 p p Find EX Solution : Example 4.2 The indicator function of an event A , denoted by I A or I ( A ) , is defined by I A = 1 if A occurs if A doesn’t occur Find E ( I A ) . Solution : 42 Example 4.3 Chuckaluck is a popular game in which three fair dice are rolled. A bet placed on one of the numbers 1 through 6 has a payoff that depends on the number of times the number appears on the three dice. Suppose one bets $ 1 on the number 4. Let X be the payoff. Then X equals  $ 1 if none of the three dice shows a 4; otherwise X = $...
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 Fall '09
 lisiufeng
 Math, Statistics

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