hw8sol_fall03 - AMS 310.01 FALL 2003 Homework#8 Solutions...

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AMS 310.01  FALL 2003   Homework #8  Solutions  2.48 Let  i i x cu a = + . Then 1 1 1 1 ( ) n n n i i i i i i x u na x cu a c cu a n n n n = = = = = + = + = + Now, 2 2 2 2 2 2 2 2 1 1 1 1 ( ) ( ) ( ) ( ) ( 1) ( 1) ( 1) ( 1) n n n n i i i i x u i i i i x x cu a cu a cu cu u u s c c s n n n n = = = = - + - - - - = = = = = - - - - Thus,  x u s c s = 5.72 (a) The independent random variables  1 2 and X X have the same probability distribution  ( ;2,0.3) b x Hence the joint probability distribution of  1 2 and X X  is 1 2 1 2 ( , ) ( ;2,0.3) ( ;2,0.3) f x x b x b x =                 1 1 2 2 1 2 1 2 2 2 4 1 2 1 2 2 2 2 2 (0.3) (0.7) (0.3) (0.7) (0.3) (0.7) x x x x x x x x x x x x - - + - - � � � � � �� � = = � � � � � �� � � � � � � �� � where  1 2 0,1,2, and 0,1,2. x x = = (b)  1 2 ( ) (0,1) (0,2) (1,2) P X X f f f < = + +                           1 3 2 2 3 1 2 2 2 2 2 2 (0.3) (0.7) (0.3) (0.7) (0.3) (0.7) 0 1 0 2 1 2 ���� ���� ���� = + + ���� ���� ���� ���� ���� ����                          = 0.2058 + 0.0441 + 0.0378  =  0.2877   
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