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Unformatted text preview: Solution for 36217 Wanjie Wang Teacher: Jiashun Jin October 7, 2009 If there is any question, please contact me at wwang@stat.cmu.edu 1. (10 Points) Calculate E ( X  Y = i ) for i =1,2,3... (3 Points) E ( X  Y = 1) = [1 p (1 , 1) + 2 p (2 , 1) + 3 p (3 , 1)] / ( p (1 , 1) + p (2 , 1) + p (3 , 1)) = 2 (3 Points) E ( X  Y = 2) = [1 p (1 , 2) + 2 p (2 , 2) + 3 p (3 , 2)] / ( p (1 , 2) + p (2 , 2) + p (3 , 2)) = 5 / 3 (3 Points) E ( X  Y = 3) = [1 p (1 , 3) + 2 p (2 , 3) + 3 p (3 , 3)] / ( p (1 , 3) + p (2 , 3) + p (3 , 3)) = 12 / 5 (1 Point)The expectation of X is different when Y is given different. So, X and Y are dependent. 2. (10 Points) Calculate the conditional distribution of X = k given X + Y = n ... P ( X + Y = n ) = n X i =0 P ( X = i,Y = n i ) = n X i =0 P ( X = i ) P ( Y = n i ) = 2 n n ! e 2 P ( X = k  X + Y = n ) = P ( X = k,Y = n k ) P ( X + Y = n ) = e 2 /k !( n k )!...
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 Fall '09
 Jin

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