Mid-year Examinations, 2017 STAT211-17S1 (C)
No electronic/communication devices are permitted.
No exam materials may be removed from the exam room.
Mathematics and Statistics
EXAMINATION
Mid-year Examinations, 2017
STAT211-17S1 (C) Random Processes
Examination Duration: 150 minutes
Exam Conditions:
Restricted Book exam: Students may bring in to the exam materials that the examiner
permits.
Any scientific/graphics/basic calculator is permitted.
Materials Permitted in the Exam Venue:
Restricted Book exam materials.
One A4 single-sided, hand-written, page of notes.
Materials to be Supplied to Students:
1 x Write-on question paper/answer book
Instructions to Students:
Use blue or black ink only (not pencil)
Attempt ALL questions (11)
There is a total of 75 marks
Show ALL working
If you use additional paper this must be tucked inside the question/answer book and you
must write your full name and your student number on every side of every page.
Family Name _____________________
First Name _____________________
Student Number |__|__|__|__|__|__|__|__|
Venue ____________________
Seat Number ________
For Examiner Use Only
Question Mark
Total ________
Mid-year Examinations, 2017 STAT211-17S1 (C)
Questions Start on Page 3
Page 2 of 23
Mid-year Examinations, 2017 STAT211-17S1 (C)
Q1 [6 marks]
Let X be a continuous random variable with the following PDF
f(x) =
cx2 00.
(a) Find c such that f(x) is a valid PDF. [2 marks]
(b) Find P(1 12 [2 marks]
Page 12 of 23
Mid-year Examinations, 2017 STAT211-17S1 (C)
Q6 [8 marks]
Find the period of each state in the following Markov chains, assuming that all drawn transitions
on the states f1,2,3,4,5g. [2 marks]
(b) Identify the communicating classes of the Markov chain and identify whether they are open
or closed. Write them in set notation and mark them on the transition diagram. [2 marks]
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Mid-year Examinations, 2017 STAT211-17S1 (C)
(c) Identify the positive recurrent and transient states. [2 marks]
Page 16 of 23
Mid-year Examinations, 2017 STAT211-17S1 (C)
Q8 [7 marks]
(a) Define the properties of a homogeneous Poisson process. Include a conceptual diagram
illustrating the process, and label the diagram to support your answer. [3 marks]
(b) How is an inhomogeneous Poisson process different? [1 marks]
Page 17 of 23
Mid-year Examinations, 2017 STAT211-17S1 (C)
(c) Describe how the following distributions are related to a homogeneous Poisson process
with rate .
Poisson( ) f(x) = xexp- x! x> 0
Exponential( ) f(x) = exp- t x> 0
Gamma(n, ) f(x) = exp- t ( t)n-1(n-1)! x> 0
Your description should identify what each distribution is used to model, and should include
at least one conceptual diagram similar to the one used in part (a). [3 marks]
Page 18 of 23
Mid-year Examinations, 2017 STAT211-17S1 (C)
Q9 [10 marks]
Parking spaces near the university are taken almost immediately during the day, as they become
available. The event of a parking space becoming available follows a Poisson process with rate
= 1 space per 3 minutes.
(a) What is the expected value and variance for the number of parking spaces becoming
available in an hour? [2 marks]
(b) What is the expected waiting time for a space to become available? [1 marks]
(c) A student has been waiting for a space for 2 minutes already. Find the probability that they
will have to wait for at least 10 minutes in total. [2 marks]
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Mid-year Examinations, 2017 STAT211-17S1 (C)
(d) What is the probability that exactly 15 spaces become available in one hour? [2 marks]
(e) You and your friend are driving to university. Find the probability that you will wait between
4 and 8 minutes for a space, given that your friend found a space in 3 minutes. [3 marks]
Page 20 of 23
Mid-year Examinations, 2017 STAT211-17S1 (C)
Q10 [5 marks]
Let fN(t) : t> 0g be a Poisson process with rate > 0, and suppose t0 = 0 are the successive occurrence times of events in the process. Prove that the interarrival times
sn =tn-tn-1 are independent and identically distributed according to Exponential( ).
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Mid-year Examinations, 2017 STAT211-17S1 (C)
Q11 [5 marks]
The purchase of coffee from a cafe on campus follows a Poisson process with rate = 300
coffees per day. The price Y of each coffee purchased is uniformly distributed on f3.5,4.5,5.0g.
The total amount of money the cafe makes in time t is defined
S(t) =N(t)Xi=1Yi
where Y1,...,YN(t) are iid according to Y.
Recall
E[S(t)] =E[N(t)]E[Y]
VAR[S(t)] =E[N(t)]VAR[Y]+VAR[N(t)]E[Y]2
(a) What is the expected amount of money the cafe will make in one week? [2 marks]
(b) What is the variance of the amount of money the cafe will make in one day? [2 marks]
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Mid-year Examinations, 2017 STAT211-17S1 (C)
(c) Would these results change for a Poisson process where (t) depends on the time of day
as well? [1 marks]
END OF EXAMINATION