课程名称︰ 高等计算机网路 课程性质︰ 选修课程 课程教师︰ 逄爱君 开课学院： 电机资讯学院 开课系所︰ 资讯工程研究所 考试日期（年月日）︰ 100/01/12 考试时限（分钟）： 180 是否需发放奖励金： 是 （如未明确表示，则不予发放） 试题 : 1. (10%) let Sn = x1 + x2 + x3 + ... + xn, where n >=1 and xi are i.i.d random variables. then Var[Sn] = ? (Hint: Var[x] = E[Var[x|y]] + Var[E[x|y]]) 2. (8%) please briefly describe "independent increment" and "stationary increment". 3. (15%) in good years, storms occur according to a Poisson process with rate 3 per unit time, while in other years they occur according to a Poisson process with rate 5 per unit time. suppose next year will be a good year with probability 0.3. let N(t) denote the number of storms during the first t time units of next year. A. find P{N(t) = n}. B. is {N(t)} a Poisson process? C. does {N(t)} have stationary increments? why or why not? D. does it have independent increments? why or why not? E. if next year starts off with three storms by time t=1, what is the conditional probability it is a good year? 4.(10%) probability inequality A. if the expected response time of a computer system is 1 second, please give the intuition based on simple Markov's inequality. (Hint: P(x >= t) <= E[x]/t , t>0 ) B. derive the tightest Chernoff's Bound for a Poisson random variable x. (Hint: P(x >= a) <= exp(-ta)*M(t) ) , for all t>0 and M(t) = exp(u*(exp(t)-1)) ) 5.(10%) is it true that A. {n(t) < n} if and only if {Sn >= t}? B. {n(t) <=n} if and only if {Sn >= t}? C. {n(t) >=n} if and only if {Sn <= t}? please justify your answer. 6.(5%) what does PASTA say? 7.(6%) please give the intuition (in your own words) of Conditional Distribution of the Arrival times" for Poisson process. also elaborate on "order statistics". 8.(6%) please briefly describe superposition and decomposition of Poisson process. 9.(10%) suppose that during a thunderstorm, shocks occur according to a Poisson process with rate 15, and suppose that each shock, independently, causes the system to fail with probability 0.01. let N denote the number of shocks that it takes for the system to fail and let T denote the time of failure. find P(N=10|T=2). 10.(10%) an insureance company pays out claims on its life insureance policies in accordance with a Poisson process having rate lamda = 4 per week. if the amount of money paid on each policy is exponentially distributed with mean $5000, what is the mean and variance of the amount of money paid by the insurance company in a five-week span? 11.(10%) events occur according to a nonhomogeneous Poisson process whose integrated intensity function is given by m(t) = t^2 + 2t what is the probability that n events occur between times t=4 and t=5? --

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