312-midterm2sol

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

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Stat312: Midterm I Solution Moo K. Chung, Yulin Zhang mchung@stat.wisc.edu, yulin@stat.wisc.edu March 5, 2003 1. Let X 1 , ··· , X n be a random sample from Bernoulli distribution with parameter p . (a) What is ( 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 ( S 2 /p 2 ) = ( 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 ( 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
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This homework help was uploaded on 01/31/2008 for the course STAT 312 taught by Professor Chung during the Spring '04 term at Wisconsin.

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