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UC BerkeleyDepartment of Electrical Engineering and Computer SciencesEECS 126: Probability and Random ProcessesProblem Set 2Spring 20201.VarianceIfX1, . . . , Xn, wheren2Z>0, are i.i.d. random variables with zero-mean andunit variance, compute the variance of (X1+· · ·+Xn)2. You may leave youranswer in terms ofE[X41], which is assumed to be finite.
2.Message SegmentationThe number of bytesNin a message has a geometric distribution with parameterp.Suppose that the message is segmented into packets, with each packetcontainingmbytes if possible, and any remaining bytes being put in the lastpacket. LetQdenote the number of full packets in the message, and letRdenote the nubmer of bytes left over.1
(a)Find the joint PMF ofQandR. Pay attention on the support of thejoint PMF.(b) Find the marginal PMFs ofQandR.(c) Repeat part (b), given that we know thatN > m.Note:you can use the formulasnXk=0ak=1-an+11-a,fora6= 11Xk=0xk=11-x,for|x|<1in order to simplify your answer.
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=p(1-p)m-11Xq=0((1-p)m)q=p(1-p)m-11-(1-p)m.(c) Due to the memoryless property of the geometric distribution, the PMF

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Term
Fall
Professor
ee126
Tags
Probability theory