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homework3_452f10 - Math 452/550 Stochastic processes...

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Math 452/550: Stochastic processes Homework assignment # 3 Due In Class, Nov. 9, 2010 Submit all of your work for marking. You are permitted to collaborate on the homework; however, you must write up your homework yourself. You should not copy somebody else’s homework: if you choose to collaborate, you should be able to recreate all of the steps involved in solving a problem yourself, and should do so in your writeup. Please list the names of your collaborators on the first page of each homework. Assignments must be submitted by the given due date. No late homework will be accepted. 1. Let X 0 be an exponentially distributed random variable. (a) Show that E [ X 2 /X > 1] = E [( X + 1) 2 ] . (b) Deduce the result in (a) without any computation, i.e, by using simple the properties of the exponential distribution. (c) Is the result valid if X is not exponentially distributed? Justify you answer with a counter- example. 2. Let X 1 , X 2 , X 3 be three exponentially distributed random variables with rates λ 1 , λ 2 , λ 3 , respec- tively. (a)Show that P { X 1 < X 2 < X 3 } = λ 1 λ 1 + λ 2 + λ 3 λ 2 λ 2 + λ 3 .
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