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hw6-sol - AMS 311(Fall 2009 Joe Mitchell PROBABILITY THEORY...

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AMS 311 (Fall, 2009) Joe Mitchell PROBABILITY THEORY Homework Set # 6 – Solution Notes (1). (20 points) [See related problems 39-41, Ross (Chap 6). See Examples 1a, 1b.] Suppose that X and Y have joint mass function as shown in the table below. (Here, X takes on possible values in the set {− 2 , 1 , 3 } , Y takes on values in the set {− 2 , 0 , 1 , 3 . 1 } .) -2 0 1 3.1 -2 .1 .2 0 0 1 .1 0 .4 0 3 0 .1 0 .1 (a). (7 points) Compute P ( | X 2 Y | > 0) . We compute P ( | X 2 Y | > 0) by summing up the values of p ( x, y ) for all of those cases for which | x 2 y | > 0. We see that the only case for which this is NOT true is (1,1), so the total is 1 p (1 , 1) = 0 . 6 (b). (7 points) Find the marginal mass function of X and plot it. (be very explicit!) Sum the rows of the table to get the marginal mass function of X : p X ( x ) = 0 . 3 if x = 2 0 . 5 if x = 1 0 . 2 if x = 3 0 otherwise (c). (6 points) Compute var ( X 2 ) var ( X 2 ) = E ( X 4 ) [ E ( X 2 )] 2 = ( 2) 4 ( . 3) + 1 4 ( . 5) + 3 4 ( . 2) [( 2) 2 ( . 3) + 1 2 ( . 5) + 3 2 ( . 2)] 2 (2). (20 points) Joe and Estie plan to study together for the AMS311 test. They decide to meet at the library between 8:00pm and 8:30pm. Assume that they each arrive at a random time in this interval. What is the probability that someone (Joe or Estie, whichever arrives first) must wait at least 12 minutes until the other one arrives? (Formulate the problem and solve. Make sure you carefully define any random variables you use!) Let X be the number of minutes past 8:00pm that Joe arrives in the library. Let Y be the number of minutes past 8:00pm that Estie arrives in the library. Then, we know that X is Uniform(0,30), and Y is Uniform(0,30). We assume that X and Y are independent; then we know that the joint density is given by f ( x, y ) = braceleftBig 1 / 900 if 0 x 30, 0 y 30 0 otherwise We want to compute P ( | X Y | > 12) = P ( X Y > 12 or X Y < 12) = integraldisplay 18 0 integraldisplay 30 x +12 (1 / 900) dydx +
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