CS 4540, Advanced AlgorithmsHomework 2Fri, Sept 10, 2010Due Fri, Sept 17, 2010Problem 1Motwani and Raghavan, Problem 4.1, page 97. Note:The purpose of this problem is to familiarizeyou with the use of Chernoff bounds.You may use any of the foffowing forms of Theorems 1through 5 given at the Class Notes of 09-08-10 (see Class Notes on the class web site).Problem 2Let 0≤y1≤y2≤. . .≤y2N≤1 be 2Nreal numbers, withy2N+1-j= 1-yj, for 1≤j≤N.LetX1, X2, . . . , Xnbe independent random variables such that, for 1≤i≤nand for 1≤j≤N,Pr [Xi=yj] =pijand Pr [Xi=y2N+1-j] = Pr [Xi=yj].Letpi=E[Xi] =∑2Nj=1yjpij, where0<pi<1. Finally, letX=∑ni=1Xnand letμ=E[X].(a) Prove that, for anyδ >0, Pr [X >(1 +δ)μ]<heδ(1+δ)1+δiμ.(b) Prove that, for any 1> δ >0, Pr [X <(1-δ)μ]<he-δ(1-δ)1-δiμ.Note:The purpose of this problem is to familiarize you with the proof of the basic form of Chernoffbounds. You may use as guideline the proof of (13.42) in page 749 of Kleinberg and Tardos, and/orthe proofs of Theorems 4.1 and 4.2 in pages 68 and 70 (respectively) of Motwani and Raghavan.
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