Student number:
The University of Melbourne
Semester 2 Assessment, 20xx
School of Mathematics and Statistics
MAST90019 Random Processes
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Instructions to Students
• This paper has eight (8) questions.
• Attempt as many questions, or parts of questions, as you can.
• All answers will be marked and credit given for all parts of answers.
• Working and/or reasoning must be given to obtain full credit.
• The marks for each question are given after the question number, and there is a
total of 115 marks on the paper.
• If you have a printer, print the exam. If you cannot print, download the exam to a
second device, which must then be disconnected from the internet.
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with supervisor permission. The screen of any other device must be visible in Zoom
from the start of the session.
• Write your answers on A4 paper. The first page should contain only your student
number, the subject code and the subject name. Write on one side of each sheet
only. Start each question on a new page and include the question number at the
top of each page.
• Assemble your single-sided solution pages in correct order and the correct way
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clearly readable and cropped to the A4 borders of the original page. Poorly scanned
submissions may be impossible to mark.
• Submit your PDF file to the Canvas Assignment corresponding to this exam using
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Page 1 of 4
Question 1 (5 marks) Give a definition of the separability of a random process {Xt :
t ∈ T} on a probability space (Ω,F ,P).
Question 2 (20 marks) Let ξ and η be independent standard normal random variables,
Xt = ξ
√
t+ η
√
1− t, 0 ≤ t ≤ 1.
(a) For a fixed t > 0, find the distribution of Xt.
(b) Find the probability P(the trajectory of Xt, 0 ≤ t ≤ 1, is above 0).
(c) Calculate Cov (Xs, Xt), 0 ≤ s < t.
(d) For t > 0, find the conditional distribution of Xt given X0 = x.
(e) Find the joint cumulative distribution function Ft1,t2,...,tn(x1, x2, . . . , xn) of the pro-
cess {Xt}t≥0 (assuming that 0 < t1 < t2 < · · · < tn, all xj ∈ R).
Question 3 (12 marks) Let X1, . . . , Xn be independent and identically distributed
integer-valued random variables, with P(X1 = 0) = p ∈ (0, 1). Define
Nn =
n−2∑
i=1
I{Xi=Xi+1=Xi+2=0}.
That is, Nn is the number of times the pattern 0, 0, 0 appears in the sequence.
(a) Define an appropriate filtration {Fi : 0 ≤ i ≤ n} such that Mi := E(Nn|Fi),
0 ≤ i ≤ n, forms a martingale with M0 = ENn and Mn = Nn.
(b) Hoeffding’s inequality states that if {(Yi,Fi) : i ≥ 0} is a martingale and there
exists a sequence of real numbers K1, K2, . . . such that |Yi− Yi−1| ≤ Ki a.s. for all
i, then
P(|Yn − Y0| ≥ x) ≤ 2 exp
{
− x
2
2
∑n
i=1K
2
i
}
, x > 0.
Use Hoeffding’s inequality to establish a bound for
P(|Nn − ENn| ≥ x), x > 0.
Question 4 (15 marks) Let {Xt}t≥0 be a drop-shot process defined by
Xt :=
∑
j≥1
e−(t−Tj)1[Tj ,∞)(t) =
Nt∑
j=1
e−(t−Tj),
where {Tj}j≥1 are the times of successive jumps in a Poisson process {Nt, t ≥ 0} with
rate λ = 1.
(a) Verify that Xt = e
−(t−s)Xs +
∑Nt
j=Ns+1
e−(t−Tj) for all t > s > 0.
(b) Argue that {Xt, t ≥ 0} is a Markov process and find its generator.
(c) Let F be the stationary distribution of the Markov process and define ψ(u) =∫
R e
−uxdF (x), u ≥ 0, as the Laplace transform of F . Solve for ψ(u) and simplify
your answer as much as you can.
Page 2 of 4
Question 5 (20 marks) Let {ξi : i ≥ 1} be a sequence of independent and identically
distributed random variables with distribution P(ξ1 = −1) = P(ξ1 = 1) = 12 . Define
S0 = a and Sn = a +
∑n
i=1 ξi for i ≥ 1. Assume a,K are positive integers with a < K.
Let T be the first time that the random walk {Sn : n ≥ 0} hits 0 or K.
(a) Define an appropriate filtration {Fn : n ≥ 0} such that {(Sn,Fn) : n ≥ 0} forms
a martingale.
(b) Explain why T is a stopping time with respect to the filtration {Fn}.
(c) Show that {(Sn∧T ,Fn) : n ≥ 0} is a uniformly integrable martingale and deduce
P(ST = K) = a/K.
(d) Set Xn = S
2
n − n, n ≥ 0. Prove that {(Xn,Fn) : n ≥ 0} is a martingale and
ET <∞.
(e) Let Mn =
∑n
j=0 Sj − 13S3n, n ≥ 0. Show that {(Mn,Fn) : n ≥ 0} is a martingale.
(f) Deduce that
E
(
T∑
j=0
Sj
)
=
1
3
(K2 − a2)a+ a.
Question 6 (8 marks) Let F be the distribution of a Bernoulli random variable with
parameter p ∈ (0, 1). Show that F is NOT infinitely divisible.
Question 7 (15 marks) Let {Xt} be a Le´vy process, such that in the Le´vy–Khintchin
representation for its characteristic function φX1 one has
ψ(u) = lnφX1(u) = iua+
∫ (
eiux − 1− iux
1 + x2
)
1 + x2
x2
dH(x),
with a = 0 and
H(x) =

0, x < −2;
1, −2 ≤ x < 0;
2(1 + x− arctanx), 0 ≤ x < 1;
4− pi/2, x ≥ 1.
(a) Give a representation of the process as a sum of a linear drift and independent
compound Possion process with negative jumps, compound Possion process with
positive jumps, and a (not necessarily standard) Wiener process. Specify the pa-
rameters of all three processes (i.e. the rates and compounding distributions for the
first two and the variance for the third one).
(b) Find the mean function mt = EXt and variance Var (Xt) of the process.
Page 3 of 4
Question 8 (20 marks) Flaws in sheet metal occur as a Poisson process with intensity
0.02/cm2 and 10% of the flaws are classified as unrepairable. A technician inspects a
strip of the sheet metal which is 10 cm long and 5 cm wide.
(a) Find the probability that it has exactly 2 flaws.
(b) Given the strip has exactly 2 flaws, what is the probability that they both occur in
a given strip 10 cm long and 1 cm wide?
(c) Given that the strip has exactly two repairable flaws, what is the probability that
these two are the only flaws in the strip?
(d) Find the probability that the distance from the lower left corner to the nearest flaw
is ≤ 2 cm.
End of examination Page 4 of 4 total marks = 115

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