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

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Stat312: Sample Midterm IIMoo K. ChungSeptember 30, 20041. LetX1,· · ·, Xnbe a random sample from Bernoullidistribution with parameterp.(a) What isE(S2/p2)?S2is the sample variance. Ex-plain your results (10 points).(b) Find an unbiased estimator ofp2.Explain yourresults (5 points).Solution.(a) The sample variance is an unbiased esti-mator of the population variance. HenceE(S2/p2) =E(S2)/p2=Var(Xi)/p2.The variance for a Bernoullirandom variable can be esily computed asVar(Xi) =p(1-p). SoE(S2/p2) = (1-p)/p. (b) We knowS2and¯Xwill be unbiased estimators of population variancep(1-p)and meanprespecively. SoE(S2)-E(¯X) =p(1-p)-p=-p2. Hence,¯X-S2is an unbiasedestimator ofp2.2. LetX1, X2be a random sample fromN(0,1). Notethat the sample size is 2 and the density function forXiisf(xi) =θ2πexp(-θx2i/2).(a) Obtain an estimator ofθusing the method of mo-ments. Explain your results (5 points).
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Term
Fall
Professor
Chung
Tags
Probability theory, Estimation theory
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