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
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• 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
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• 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.
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©University of Melbourne 2021 Page 1 of 16 pages Can be placed in Baillieu Library
Student number
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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.
ShowthatifW ∼Exp(λ)withλ>0thenLW(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.
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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: 1/3
(b) Suppose that P(X0 = 1) = 1. Find P(X4 = 1).
(c) Determine the communicating classes of the chain.
(a) Determine the value of c and find the one-step transition matrix for the chain.
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MAST30001 Stochastic Modelling Semester 2, 2021
(d) Is this chain periodic or aperiodic? If periodic, specify the period.
(e) SupposethatP(X0 =1)=1,andletT ≥0bethetimeofthelastvisittostate1before 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.
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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) Explainwhyh0,−1 <1if2b+3a>1.
(c) Explainwhyh0,−1 =1if2b+3a<1.
(d) Show that h0,−1 satisfies a cubic equation that has 1 as a solution.
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MAST30001 Stochastic Modelling
Semester 2, 2021
(e) Find h0,−1 when a = b = 1/4.
(f) Findh0,−1 when2b+3a=1.
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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
a10000 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.
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MAST30001 Stochastic Modelling
Semester 2, 2021
(d) Let T = inf{t > 0 : Xt ̸= 1}. Find the distribution of T.
(e) Is T a stopping time? Why or why not?
(f) LetT′ =inf{t>0:Xt =6}. FindtheexpectedvalueofT′.
(g) Evaluate the Laplace transform of T′.
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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.
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MAST30001 Stochastic Modelling
Semester 2, 2021
Question 6 (12 marks)
A CTMC (Xt)t≥0 with the following transition diagram λ
startsinstate0,i.e.P(X0 =0)=1. LetTi =inf{t≥0:Xt =i}fori∈{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?
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MAST30001 Stochastic Modelling Semester 2, 2021
(e) Give an infinite series expression for the quantity P (t) for each i ∈ {0, 1, . . . , 5}. 0,i
(f) Show that your answers to part (e) satisfy the Kolmogorov forward equations for P(t), 0,i
i = 0,1,…,5.
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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,….
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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)2W2].
(b)LetT1 =inf{t≥0:Wt =1}andT±1 =inf{t≥0:|Wt|=1}. FindE[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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