ECE 534 RANDOM PROCESSES FALL 2008
PROBLEM SET 7 Due Wednesday, December 10
7. Random Processes in Linear Systems and Spectral Analysis
Assigned Reading: Sections 8.1-8.4 and 9.1 of the notes.
Reminder: The final exam will be on Monday, December 15, 1:30-4:30 p.m., Room 66 in
the Main Library, which is the large building just south of Gregory Hall. Room 66 is located
in the northwest basement, accessed by the west elevator. There is also outside access by
stairs near the north library door.
Problems to be handed in:
1 Some linear transformations of some random processes
Let U= (Un:n∈Z) be a random process such that the variables Unare independent,
identically distributed, with E[Un] = µand Var(Un) = σ2, where µ6= 0 and σ2>0. Please
keep in mind that µ6= 0. Let X= (Xn:n∈Z) be defined by Xn=P∞
k=0 Un−kak, for a
constant awith 0 < a < 1.
(a) Is Xstationary? Find the mean function µXand autocovariance function CXfor X.
(b) Is Xa Markov process ? (Hint: Xis not necessarily Gaussian. Does Xhave a state
representation driven by U?)
(c) Is Xmean ergodic in the m.s. sense?
Let Ube as before, and let Y= (Yn:n∈Z) be defined by Yn=P∞
k=0 Un−kAk, where Ais a
random variable distributed on the interval (0,0.5) (the exact distribution is not specified),
and Ais independent of the random process U.
(d) Is Ystationary? Find the mean function µYand autocovariance function CYfor Y.
(Your answer may include expectations involving A.)
(e) Is Ya Markov process? (Give a brief explanation.)
(f) Is Ymean ergodic in the m.s. sense?
2 Causal prediction of a Poisson process
Let N= (Nt:t≥0) be a Poisson process with rate λ > 0.
(a) Let S= min{t≥0 : Nt≥3}, which is the time of the third jump of N.Write down the
probability density of S, being as explicit as possible.
(b) Find b
E[N20|Ns,0≤s≤10], which is defined to be the minimum mean square error
linear estimator of N20 given (Ns: 0 ≤s≤10). (Hint: No calculation is necessary.)
3 On an M/D/infinity system
Suppose customers enter a service system according to a Poisson point process on Rof rate λ,
meaning that the number of arrivals, N(a, b],in an interval (a, b], has the Poisson distribution
with mean λ(b−a), and the numbers of arrivals in disjoint intervals are independent. Suppose
each customer stays in the system for one unit of time, independently of other customers.
Because the arrival process is memoryless, because the service times are deterministic, and
because the customers are served simultaneously, corresponding to infinitely many servers,
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