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Practice Exams and Their Solutions Based on
Practice Exams and Their SolutionsBased onA Course in Probability and StatisticsCopyrightc 2003 5 by Charles J. StoneDepartment of StatisticsUniversity of California, BerkeleyBerkeley, CA 94720-3860 Please email corrections and other comments to (Chapters 1 6) Practice ExamsFirst Practice First Midterm Exam1. Write an essay on variance and standard LetWhave the exponential distribution with mean 1. Explain howWcanbe used to construct a random variableY=g(W) such thatYis uniformlydistributed on{0,1,2}.3. LetWhave the density functionfgiven byf(w) = 2/w3forw >1 andf(w) = 0 forw 1. SetY= + W, where >0. In terms of and ,determine(a) the distribution function ofY;(b) the density function ofY;(c) the quantiles ofY;(d) the mean ofY;(e) the variance LetYbe a random variable having mean and suppose thatE[(Y )4] this information to determine a good upper bound toP(|Y | 10).
Hint: Consider the power series expansion ... Determine a formula for the joint density function of Y 1, Y 2, and Y 3 in terms of the joint density function of W 1, W 2, and W 3. (b) Suppose that W 1, W 2, and W 3 are independent random variables, each
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