# HW4 sol.pdf - Homework 4 Math 561: Theory of Probability I...

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Homework 4Math 561: Theory of Probability IDue date: February 25, 2021Each problem is worth 10 points and only five randomly chosen problems will be graded if there are more than5 problems. Please indicate whom you worked with, it will not affect your grade in any way.1. (i) Prove that Markov’s inequalityP(X>t)6E(X)/tis sharp for fixedt >0,i.e.,there is a non-negativerandom variableXsuch thatP(X>t) =E(X)/t.(ii) (One-sided Chebyshev inequality) Suppose thatE(X) = 0, Var(X) =σ2<anda >0. Prove thatP(X>a)6σ2σ2+a2and there is a r.v.Xfor which equality holds.

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Term
Fall
Professor
Laugesen
Tags
Probability theory, E X, Markov s inequality P