hw4 fall 07 solns

hw4 fall 07 solns - 1 ISyE 3770 Fall 2007 Homework #4...

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Unformatted text preview: 1 ISyE 3770 Fall 2007 Homework #4 (Covers Modules 1415) Solutions 1. Suppose that X is a discrete random variable with p.m.f. Pr ( X =- 1) = 0 . 5, Pr ( X = 0) = 0 . 2, and Pr ( X = 2) = 0 . 3. Find E [ X ], Var ( X ), and E [ X 3 ]. Solution: By definition of expectation and variance for discrete random variables: E [ X ] = Pr ( X =- 1) (- 1) + Pr ( X = 0) 0 + Pr ( X = 2) 2 =- . 5 + 0 + 0 . 6 = 0 . 1 Var [ X ] = E [ X 2 ]- E [ X ] 2 = 0 . 5 (- 1) 2 + 0 . 2 (0) 2 + 0 . 3 (2) 2- (0 . 1) 2 = 1 . 69 E [ X 3 ] = Pr ( X =- 1) (- 1) 3 + Pr ( X = 0) (0) 3 + Pr ( X = 2) (2) 3 =- . 5 + 0 + 2 . 4 = 1 . 9 . 2. Suppose that X is a continuous random variable with p.d.f. f ( x ) = cx for 0 < x < 1. Find c , E [ X ], Var ( X ), and E [ X 3 ]. Solution: Use the fact that the p.d.f. of a random variable, integrated over its domain, equals one: Z 1 cxdx = c 2 = 1 ....
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hw4 fall 07 solns - 1 ISyE 3770 Fall 2007 Homework #4...

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