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Unformatted text preview: ¯ X n is the sample mean, that is, ¯ X n = 1 n ∑ n i =1 X i . Show that the random variable S 2 n , which is called the sample variance, has expectation σ 2 . In statistical terminology, S 2 n is said to be an unbiased estimator of σ 2 . Extra Credit (3 points) 6. (Continuation of problem 5) Using the notation and assumptions of problem [5], assume further that the third central moment of the distribution of X equals zero. Prove that the sample mean and the sample variance are uncorrelated, using the following steps: (a) Let Y j = X jμ X for 1 ≤ j ≤ n , and let T 2 n denote the sample variance of the Y j ’s. Show that Cov ( ¯ X n , S 2 n ) = Cov ( ¯ Y n , T 2 n ) . (b) Prove that Cov ( ¯ Y n , T 2 n ) = 0 . (c) Conclude that ¯ X n and S 2 n are uncorrelated. 2...
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 Fall '11
 Wierman
 Probability, Variance, Probability theory, Tn, sample variance, 1 j

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