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Unformatted text preview: Pages 375377 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 ) = X i X j Cov ( X i Y j ) = X i Cov ( X i ,Y i ) = n 1 36 The above used Cov ( X i Y j ) = 0, whenever i 6 = 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 X i =1 V i !...
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This note was uploaded on 01/23/2011 for the course STAT 418 at Pennsylvania State University, University Park.
 '08
 G.JOGESHBABU
 Probability

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