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Ex 1.4: Suppose n observations X1, X2,..., Xn have mean μ, variance σ², and cor(Xi, Xi+1) = p for all i. All other correlations equal zero. (a) If show that X = n n Σχε var (X) = (1 + n 2(n−1)p). (b) Deduce that if p > 0 the confidence interval for μ will be wider than in the case with {X+} independent. What happens if p < 0? Hints: (1) Recall that n n var(X) = ΣΣ cov(Xi, Xj). i=1 j=1 How many terms have i = j? How many have i - j = ±1? (2) Recall that cov(Xi, X;) = cor(Xi, X;)√/var(X;) var(X;).

Ex 1.4: Suppose n observations X1, X2,..., Xn have mean μ, variance σ², and cor(Xi, Xi+1) = p for all i. All other correlations equal zero. (a) If show that X = n n Σχε var (X) = (1 + n 2(n−1)p). (b) Deduce that if p > 0 the confidence interval for μ will be wider than in the case with {X+} independent. What happens if p < 0? Hints: (1) Recall that n n var(X) = ΣΣ cov(Xi, Xj). i=1 j=1 How many terms have i = j? How many have i - j = ±1? (2) Recall that cov(Xi, X;) = cor(Xi, X;)√/var(X;) var(X;).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Ex 1.4: Suppose n observations X1, X2,..., Xn have mean μ, variance σ², and cor(Xi, Xi+1) = p for all i. All other correlations equal zero. (a) If show that X = n n Σχε var (X) = (1 + n 2(n−1)p). (b) Deduce that if p > 0 the confidence interval for μ will be wider than in the case with {X+} independent. What happens if p < 0? Hints: (1) Recall that n n var(X) = ΣΣ cov(Xi, Xj). i=1 j=1 How many terms have i = j? How many have i - j = ±1? (2) Recall that cov(Xi, X;) = cor(Xi, X;)√/var(X;) var(X;).
Transcribed Image Text:Ex 1.4: Suppose n observations X1, X2,..., Xn have mean μ, variance σ², and cor(Xi, Xi+1) = p for all i. All other correlations equal zero. (a) If show that X = n n Σχε var (X) = (1 + n 2(n−1)p). (b) Deduce that if p > 0 the confidence interval for μ will be wider than in the case with {X+} independent. What happens if p < 0? Hints: (1) Recall that n n var(X) = ΣΣ cov(Xi, Xj). i=1 j=1 How many terms have i = j? How many have i - j = ±1? (2) Recall that cov(Xi, X;) = cor(Xi, X;)√/var(X;) var(X;).
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