Unformatted text preview: y j π j ! 2 = 1 2 X i ∈ s X j ∈ s π i π jπ ij π ij y i π iy j π j ! 2 (5) Techniques useful for proofs: i) Let a i = 1 if i ∈ s and 0 otherwise, i = 1 , 2 , ··· ,N . Then E ( a i ) = π i , E ( a i a j ) = π ij for i 6 = j , V ( a i ) = π i (1π i ), and Cov ( a i ,a j ) = π ijπ i π j . ii) To ﬁnd out the expectation (or variance) of A = ∑ i ∈ s Z i and B = ∑ i ∈ s ∑ j ∈ s Z i Z j , we rewrite A and B as A = N X i =1 a i Z i and B = N X i =1 N X j =1 a i a j Z i Z j . iii) When the sample size n is ﬁxed, we have N X i =1 π i = n ; N X j 6 = i π ij = ( n1) π i and N X i =1 N X j 6 = i π ij = n ( n1) ....
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 Fall '09
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 Probability, Variance, Yi, ij, ij ij yi

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