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Student
number
Semester 2 Assessment, 2021
School of Mathematics and Statistics
MAST30001 Stochastic Modelling
Reading time: 30 minutes — Writing time: 3 hours — Upload time: 30 minutes
This exam consists of 16 pages (including this page) with 9 questions and 80 total marks
Permitted Materials
• This exam and/or an offline electronic PDF reader, one or more copies of the masked exam
template made available earlier, blank loose-leaf paper and a Casio FX-82 calculator.
• One double sided A4 page of notes (handwritten or printed).
• No headphones or earphones are permitted.
Instructions to Students
• Wave your hand right in front of your webcam if you wish to communicate with the
supervisor at any time (before, during or after the exam).
• You must not be out of webcam view at any time without supervisor permission.
• You must not write your answers on an iPad or other electronic device.
• Off-line PDF readers (i) must have the screen visible in Zoom; (ii) must only be used to
read exam questions (do not access other software or files); (iii) must be set in flight mode
or have both internet and Bluetooth disabled as soon as the exam paper is downloaded.
Writing
• Working and/or reasoning must be given to obtain full credit. Clarity, neatness and style
count.
• If you are writing answers on the exam or masked exam and need more space, use blank
paper. Note this in the answer box, so the marker knows.
• If you are only writing on blank A4 paper, the first page must contain only your student
number, subject code and 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.
Scanning and Submitting
• You must not leave Zoom supervision to scan your exam. Put the pages in number
order and the correct way up. Add any extra pages to the end. Use a scanning app to
scan all pages to PDF. Scan directly from above. Crop pages to A4.
• Submit your scanned exam as a single PDF file and carefully review the submission in
Gradescope. Scan again and resubmit if necessary. Do not leave Zoom supervision until
you have confirmed orally with the supervisor that you have received the Gradescope
confirmation email.
• You must not submit or resubmit after having left Zoom supervision.
©University of Melbourne 2021 Page 1 of 16 pages Can be placed in Baillieu Library
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MAST30001 Stochastic Modelling Semester 2, 2021
Question 1 (2 marks)
The Laplace transform of a non-negative random variable W is defined as
LW (s) = E[e−sW ], s ≥ 0.
Show that if W ∼ Exp(λ) with λ > 0 then LW (s) = λ/(λ+ s).
Question 2 (5 marks)
Let pn be a sequence of numbers in (0, 1) that are decreasing to 0, and let Gn ∼Geometric(pn).
(a) Determine P(pnGn > x) for each x < 0. (b) Determine P(pnGn > x) for each x ≥ 0.
(c) Show that pnGn converges in distribution (as n→∞) by taking the limit as n→∞ in
your answers above, and specify the limiting distribution.
Page 2 of 16 pages
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MAST30001 Stochastic Modelling Semester 2, 2021
Question 3 (11 marks)
A simple DTMC (Xn)n∈Z+ with state space S = {1, 2, 3} has transition diagram:
1 2
3
1/3
2/3
1/3
2/3
c
1
(a) Determine the value of c and find the one-step transition matrix for the chain.
(b) Suppose that P(X0 = 1) = 1. Find P(X4 = 1).
(c) Determine the communicating classes of the chain.
Page 3 of 16 pages
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MAST30001 Stochastic Modelling Semester 2, 2021
(d) Is this chain periodic or aperiodic? If periodic, specify the period.
(e) Suppose that P(X0 = 1) = 1, and let T ≥ 0 be the time of the last visit to state 1 before
the first visit to state 3. Is T a stopping time for this chain? Why or why not?
(f) Find the limiting distribution for this chain if P(X0 = 1) = 1.
(g) Find the limiting distribution of this chain if the initial distribution is uniform on S.
Page 4 of 16 pages
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MAST30001 Stochastic Modelling Semester 2, 2021
Question 4 (10 marks)
Let (Xn)n∈Z+ be a DTMC with state space S = Z and transition probabilities pi,i+1 = b,
pi,i+2 = a and pi,i−1 = 1− (a+ b), where 0 < a, b and a+ b < 1. Suppose that P(X0 = 0) = 1
and let h0,j be the probability of ever hitting state j (starting from state 0).
(a) Find E[X1].
(b) Explain why h0,−1 < 1 if 2b+ 3a > 1.
(c) Explain why h0,−1 = 1 if 2b+ 3a < 1. (d) Show that h0,−1 satisfies a cubic equation that has 1 as a solution. Page 5 of 16 pages P a g e 6 o f 1 6 — a d d a n y ex tr a p a g es a ft er p a g e 1 6 — P a g e 6 o f 1 6 MAST30001 Stochastic Modelling Semester 2, 2021 (e) Find h0,−1 when a = b = 1/4. (f) Find h0,−1 when 2b+ 3a = 1. Page 6 of 16 pages P a g e 7 o f 1 6 — a d d a n y ex tr a p a g es a ft er p a g e 1 6 — P a g e 7 o f 1 6 MAST30001 Stochastic Modelling Semester 2, 2021 Question 5 (18 marks) Let (Xt)t≥0 be a CTMC with state space {1, 2, 3, 4, 5, 6} and corresponding generator a 1 0 0 0 0 0 b 1/2 0 0 0 0 0 c 1/3 0 0 0 0 0 d 1/4 0 0 0 0 0 e 1/5 1/6 0 0 0 0 f , and suppose that P(X0 = 1) = 1. (a) Find the values of a, b, c, d, e, f . (b) Draw the transition diagram for this chain. (c) Find all stationary distributions for the jump chain associated to this chain, and explain why the jump chain does not have a limiting distribution. Page 7 of 16 pages P a g e 8 o f 1 6 — a d d a n y ex tr a p a g es a ft er p a g e 1 6 — P a g e 8 o f 1 6 MAST30001 Stochastic Modelling Semester 2, 2021 (d) Let T = inf{t > 0 : Xt 6= 1}. Find the distribution of T .
(e) Is T a stopping time? Why or why not?
(f) Let T ′ = inf{t > 0 : Xt = 6}. Find the expected value of T ′.
(g) Evaluate the Laplace transform of T ′.
Page 8 of 16 pages
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MAST30001 Stochastic Modelling Semester 2, 2021
(h) Find the long run proportion of time spent in state 1.
(i) Draw the transition diagram for an irreducible, aperiodic DTMC (Yn)n∈Z+ with the
same state space and stationary distribution as (Xt)t≥0, and explain why your chosen
DTMC has this property.
Page 9 of 16 pages
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MAST30001 Stochastic Modelling Semester 2, 2021
Question 6 (12 marks)
A CTMC (Xt)t≥0 with the following transition diagram
0
1 2
3
45
λ
λ
λ
λ
λ
λ
starts in state 0, i.e. P(X0 = 0) = 1. Let Ti = inf{t ≥ 0 : Xt = i} for i ∈ {0, 1, . . . , 5}.
(a) Find the generator matrix Q for this chain.
(b) Find P(T2 > 1).
(c) Find the cdf of T2, conditional on the event {T2 < t, T3 > t}.
(d) Is this chain reversible? Why or why not?
Page 10 of 16 pages
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MAST30001 Stochastic Modelling Semester 2, 2021
(e) Give an infinite series expression for the quantity P
(t)
0,i for each i ∈ {0, 1, . . . , 5}.
(f) Show that your answers to part (e) satisfy the Kolmogorov forward equations for P
(t)
0,i ,
i = 0, 1, . . . , 5.
Page 11 of 16 pages
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MAST30001 Stochastic Modelling Semester 2, 2021
Question 7 (5 marks)
At the Arrivals section of a small airport, passengers arrive as a Poisson process of rate 2
per minute and each such passenger is assigned at random (based on a fair coin toss, and
independent of the length of the queues) to one of 2 servers (that each serve the customers in
their queue in the order in which they arrived). Service times at each queue are Exponential
random variables with mean φ, in minutes, and independent of everything else. Let Nt denote
the total number of customers in this system at time t.
(a) How small does φ have to be to ensure that the queuing system is stable (i.e. that
no matter how many customers are currently in the system, the time until it becomes
empty has finite expectation)?
(b) Assuming that φ is sufficiently small so that the system is stable, find the limiting
distribution for Nt as t→∞, i.e. find limt→∞ P(Nt = n) for each n = 0, 1, 2, . . . .
Page 12 of 16 pages
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MAST30001 Stochastic Modelling Semester 2, 2021
Question 8 (7 marks)
In order to attend a large sports event at a stadium nearby, ticket holders park their cars (one
behind the other) on a 1km stretch of road that lies between 2 barriers. A meticulous parking
attendant makes sure that each car leaves exactly 1m space in front of it. As soon as a car
arrives that cannot fit (with 1m space in front of it to the next car), the parking attendant sends
that car away, puts up a “no parking” sign and forbids any further parking on the stretch of
road. Assume that arriving cars have lengths that are independent and uniformly distributed
between 2.5m and 5.5m, and that the demand for car park spaces on this road is high (so the
no parking sign will eventually be put up).
Let N1000 denote the number of cars that park in this stretch of road for the event. Use your
knowledge of renewal theory to answer the following.
(a) Give a point estimate and an approximate 95% confidence interval for N1000.
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MAST30001 Stochastic Modelling Semester 2, 2021
(b) Estimate the cdf of the amount of space left behind the last parked car when the no
parking sign is put up (you may leave your answer in integral form, but any integrand(s)
should be explicit).
(c) Suppose that the expectation of the amount of space left behind the last parked car is z
(metres). What is the (approximate) expected length of the first car that is sent away
in terms of z?
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MAST30001 Stochastic Modelling Semester 2, 2021
Question 9 (10 marks)
Let (Wt)t≥0 be a standard Brownian motion.
(a) Find E[(W3 −W2)2W 22 ].
(b) Let T1 = inf{t ≥ 0 : Wt = 1} and T±1 = inf{t ≥ 0 : |Wt| = 1}. Find E[WT1 ] and
E[WT±1 ].
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MAST30001 Stochastic Modelling Semester 2, 2021
(c) Fix a > 0, and for t ≥ 0 let Yt = a−1/2Wat. Show that (Yt)t≥0 is a standard Brownian
motion.
(d) For t ∈ [0, 1] let Zt = Wt − tW1. Show that (Zt)t∈[0,1] is a standard Brownian bridge.
End of Exam — Total Available Marks = 80
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