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hw6-sol

# 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. 10 . 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 . 3i f x = - 2 0 . 5i f x =1 0 . 2i f x =3 0 otherwise (c). (6 points) Compute var ( X 2 ) ( 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 ±rst) must wait at least 12 minutes until the other one arrives? (Formulate the problem and solve. Make sure you carefully de±ne 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

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

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