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hw3 fall 07 solns - 1 ISyE 3770 Fall 2007 Homework#3(Covers...

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1 ISyE 3770 – Fall 2007 Homework #3 (Covers Modules 10–14) — Solutions 1. A die is rolled 5 times. Let X denote the number of times that you see a 3 or 6. (a) What’s the distribution of X ? Solution: X Bin(5 , 1 / 3) . (b) Find Pr ( X = 4). Solution: Pr ( X = 4) = ˆ 5 4 ! 1 3 4 2 3 5 - 4 = 0 . 0412 . 2. Suppose X Pois(2). Find Pr ( X 3). Solution: Pr ( X 3) = 3 X k =0 e - 2 2 k /k ! = 0 . 8571 . 3. Suppose X has the following discrete distribution. x -1 0 2 3 Pr ( X = x ) 0.2c 0.3 0.2 0.1 (a) Find the value of c that will make the p.m.f. sum to 1. Solution: Note that 1 = X x Pr ( X = x ) = 0 . 2 c + 0 . 3 + 0 . 2 + 0 . 1 . This implies that c = 2. (b) Find the c.d.f. F ( x ) for all x . Solution: We have F ( x ) = Pr ( X x ) = 0 if x < - 1 0 . 4 if - 1 x < 0 0 . 7 if 0 x < 2 0 . 9 if 2 x < 3 1 . 0 if x 3 (c) Calculate E [ X ]. Solution: E [ X ] = X x x Pr ( X = x ) = 0 . 3 .
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2 (d) Calculate Var ( X ). Solution: E [ X 2 ] = X x x 2 Pr ( X = x ) = 2 . 1 . This implies that Var ( X ) = E [ X 2 ] - ( E [ X ]) 2 = 2 . 01. (e) Calculate Pr (1 X 2).
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