34 mse for l x 1 14 2 1 2 2 2 1 2 2 b

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Unformatted text preview: 159 (a) For any n Var(Sn ) = Cov(X1 + · · · + Xn , X1 + · · · + Xn ) n n = Cov(Xi , Xj ) i=1 j =1 n−1 = n Cov(Xi , Xi+1 ) + i=1 n Cov(Xi , Xi ) + i=1 Cov(Xi , Xi−1 ) i=2 ≤ (n − 1)(0.1) + n + (n − 1)(0.1) < (1.2)n. (b) Therefore, Var( S100 ) = 100 inequality, P 4.9.2 S100 100 −5 ≥ 1 Var(S100 ) ≤ 0.012. (100)2 0 0.5 ≤ (0..012 = 0.048. 5)2 Also, E S100 100 = µ = 5. Thus, by Chebychev’s Central limit theorem The following version of the central limit theorem (CLT) generalizes the DeMoivre-Laplace limit theorem, discussed in Section 3.6.3. While the DeMoivre-Laplace limit theorem pertains to sums of independent, identically distributed Bernoulli random variables, the version here pertains to sums of independent, identically distributed random variables with any distribution, so long as their mean and variance are finite. Theorem 4.9.4 (Central limit theorem) Suppose X1 , X2 , . . . are independent, identically distributed random variables, each with mean µ and variance σ 2 . Let Sn = X1 + · · · + Xn . T...
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