312-midterm2-2001 and soln - Stat312 Sample Midterm II Moo K Chung 1 Let X1 Xn be a random sample from Bernoulli distribution with parameter p(a What is

Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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Stat312: Sample Midterm II Moo K. Chung September 30, 2004 1. Let X 1 , · · · , X n be a random sample from Bernoulli distribution with parameter p . (a) What is E ( S 2 /p 2 ) ? S 2 is the sample variance. Ex- plain your results (10 points). (b) Find an unbiased estimator of p 2 . Explain your results (5 points). Solution. (a) The sample variance is an unbiased esti- mator of the population variance. Hence E ( S 2 /p 2 ) = E ( S 2 ) /p 2 = Var ( X i ) /p 2 . The variance for a Bernoulli random variable can be esily computed as Var ( X i ) = p (1 - p ) . So E ( S 2 /p 2 ) = (1 - p ) /p . (b) We know S 2 and ¯ X will be unbiased estimators of population variance p (1 - p ) and mean p respecively. So E ( S 2 ) - E ( ¯ X ) = p (1 - p ) - p = - p 2 . Hence, ¯ X - S 2 is an unbiased estimator of p 2 . 2. Let X 1 , X 2 be a random sample from N (0 , 1 ) . Note that the sample size is 2 and the density function for X i is f ( x i ) = θ 2 π exp( - θx 2 i / 2) . (a) Obtain an estimator of θ using the method of mo- ments. Explain your results (5 points).

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