soln5 - Probability and Stochastic Processes A Friendly...

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Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J. Goodman Problem Solutions : Yates and Goodman,3.1.2 3.3.2 3.4.2 3.5.2 3.6.2 3.6.3 and 3.7.3 Problem 3.1.2 On the X , Y plane, the joint PMF is y x P X , Y ( x , y ) 3 c 2 c c c c c 2 c 3 c 1 2 1 (a) To find c , we sum the PMF over all possible values of X and Y . We choose c so the sum equals one. x y P X , Y ( x , y ) = x = - 2 , 0 , 2 y = - 1 , 0 , 1 c | x + y | = 6 c + 2 c + 6 c = 14 c Thus c = 1 / 14. (b) P [ Y < X ] = P X , Y ( 0 , - 1 )+ P X , Y ( 2 , - 1 )+ P X , Y ( 2 , 0 )+ P X , Y ( 2 , 1 ) = c + c + 2 c + 3 c = 7 c = 1 / 2 (c) P [ Y > X ] = P X , Y ( - 2 , - 1 )+ P X , Y ( - 2 , 0 )+ P X , Y ( - 2 , 1 )+ P X , Y ( 0 , 1 ) = 3 c + 2 c + c + c = 7 c = 1 / 2 (d) From the sketch of P X , Y ( x , y ) given above, P [ X = Y ] = 0. (e) P [ X < 1 ] = P X , Y ( - 2 , - 1 )+ P X , Y ( - 2 , 0 )+ P X , Y ( - 2 , 1 )+ P X , Y ( 0 , - 1 )+ P X , Y ( 0 , 1 ) = 8 c = 8 / 14 1
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Problem 3.3.2 On the X , Y plane, the joint PMF is y x P X , Y ( x , y ) 3 c 2 c c c c c 2 c 3 c 1 2 1 (a) To find c , we sum the PMF over all possible values of X and Y . We choose c so the sum equals one. x y P X , Y ( x , y ) = x = - 2 , 0 , 2 y = - 1 , 0 , 1 c | x + y | = 6 c + 2 c + 6 c = 14 c Thus c = 1 / 14. (b) P [ Y < X ] = P X , Y ( 0 , - 1 )+ P X , Y ( 2 , - 1 )+ P X , Y ( 2 , 0 )+ P X , Y ( 2 , 1 ) = c + c + 2 c + 3 c = 7 c = 1 / 2 (c) P [ Y > X ] = P X , Y ( - 2 , - 1 )+ P X , Y ( - 2 , 0 )+ P X , Y ( - 2 , 1 )+ P X , Y ( 0 , 1 ) = 3 c + 2 c + c + c = 7 c = 1 / 2 (d) From the sketch of P X , Y ( x , y ) given above, P [ X = Y ] = 0. (e) P [ X < 1 ] = P X , Y ( - 2 , - 1 )+ P X , Y ( - 2 , 0 )+ P X , Y ( - 2 , 1 )+ P X , Y ( 0 , - 1 )+ P X , Y ( 0 , 1 ) = 8 c = 8 / 14 Problem 3.4.2 In Problem 3.2.2, we found that the joint PMF of X and Y was y x P X , Y ( x ,
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