課程名稱︰ 高等計算機網路
課程性質︰ 選修課程
課程教師︰ 逄愛君
開課學院： 電機資訊學院
開課系所︰ 資訊工程研究所
考試日期（年月日）︰ 100/01/12
考試時限（分鐘）： 180
是否需發放獎勵金： 是
（如未明確表示，則不予發放）
試題 :
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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