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 DeMoivreLaplace limit
theorem, discussed in Section 3.6.3. While the DeMoivreLaplace 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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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.
 Spring '08
 Zahrn
 The Land

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