312-midterm2-2001 and soln

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

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Unformatted text preview: 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 ) = 2 exp(- x 2 i / 2) ....
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This homework help was uploaded on 01/31/2008 for the course STAT 312 taught by Professor Chung during the Fall '04 term at Wisconsin.

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