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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 2 . In statistical terminology, S 2 n is said to be an unbiased estimator of 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 j 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

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