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hw11_418_10 - Pages 375-377 n 36 Set X = i=1 Xi Y = I(ith...

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Pages 375-377 36 Set X = n i =1 X i , Y = n i =1 Y i , where X i = I ( i th roll results in 1) and Y i = I ( i th roll results in 2). Then Cov ( X, Y ) = i j Cov ( X i Y j ) = i Cov ( X i , Y i ) = - n 1 36 The above used Cov ( X i Y j ) = 0, whenever i = j (due to the independence of X i and Y j ), and Cov ( X i , Y i ) = E ( X i Y j ) - E ( X i ) E ( Y j ) = 0 - E ( X i ) E ( Y j ) = - (1 / 6) 2 . 37 Cov ( X 1 + X 2 , X 1 - X 2 ) = Cov ( X 1 , X 1 ) - Cov ( X 1 , X 2 ) + Cov ( X 2 , X 1 ) - Cov ( X 2 , X 2 ) = Cov ( X 1 , X 1 ) - Cov ( X 2 , X 2 ) = V ar ( X 1 ) - V ar ( X 2 ) = 0, where X i = outcome of roll i. 55 Let V i = I ( i th duck gets hit), and N the number of ducks in a flock. Thus, the number of ducks that get hit is N i =1 V i . Use E ( Y ) = E ( E ( Y | X )) with Y = N i =1 V i and X = N to get that the expected number of ducks that are hit can be found from E N i =1 V i = E E N i =1 V i N . Since any duck is equally likely to get hit (no matter what N is) we have E N i =1 V i N = NE ( V 1 N ) Thus, we need to find E ( V 1 N ) , as a function of N , and then take the expected value of NE ( V 1 N ) with the information that N Poisson( λ = 6).
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