MTH305e
Examination – July Semester 2018
Principles of Applied Probability
Thursday, 15 November 2018
4:00 pm – 6:00 pm
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Time allowed: 2 hours
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INSTRUCTIONS TO STUDENTS:
1. This examination contains SIX (6) questions and comprises FIVE (5) printed
pages (including cover page).
2. You must answer ALL questions.
3. All answers must be written in the answer book. Marks will only be awarded if
FULL working is shown.
4. This is an open-book examination.
At the end of the examination
Please ensure that you have written your examination number on each answer book used.
Failure to do so will mean that your work cannot be identified.
If you have used more than one answer book, please tie them together with the string
provided.
THE UNIVERSITY RESERVES THE RIGHT NOT TO MARK YOUR
SCRIPT IF YOU FAIL TO FOLLOW THESE INSTRUCTIONS.
MTH305e Copyright © 2018 Singapore University of Social Sciences (SUSS)
Examination – July Semester 2018
Page 1 of 5
Answer all questions. (Total 100 marks)
Question 1
(a)
At a shared-bicycle parking lot, there are two out of n bicycles having brakes that
need servicing. Four students arrive to choose a bicycle each, at random. α is the
probability that all four bicycles chosen have good brakes; β is the probability
both the two bicycles with brake issues are chosen. It was noted that α = β.
(i)
Demonstrate that
𝑛𝑛 − 4 𝑛𝑛 − 5
��
� .
𝑛𝑛
𝑛𝑛 − 1
𝛼𝛼 = �
(ii)
(3 marks)
Formulate similar expression for β, in terms of n.
(3 marks)
(iii)
(b)
Hence solve for the number, n, of available bicycle at the parking lot.
(3 marks)
In a class of twelve unrelated students, compute the probability that
(i)
none of them has birthday falling in the month of September.
(2 marks)
(ii)
at least 2 of the students having birthday in the month of September.
(4 marks)
Question 2
The random variable, Sn, is defined as
𝑆𝑆𝑁𝑁 = 𝑋𝑋1 + 𝑋𝑋2 +. . +𝑋𝑋𝑁𝑁 ,
where Xi’s are independent random variables with identical distribution given by
𝑃𝑃[𝑋𝑋𝑖𝑖 = 0] = 𝑞𝑞 , 𝑃𝑃[𝑋𝑋𝑖𝑖 = 1] = 𝑝𝑝
and N is a random variable with a Poisson distribution having mean λ.
(a)
What is the name of the random variable, Sn?
(2 marks)
(b)
Formulate (write down) the probability generating function, GN(s), of N.
(1 mark)
(c)
Formulate (write down) the probability generating function, GX(s), of each Xi.
(2 marks)
MTH305e Copyright © 2018 Singapore University of Social Sciences (SUSS)
Examination – July Semester 2018
Page 2 of 5
(d)
Formulate the probability generating function, 𝐺𝐺𝑆𝑆𝑁𝑁 (𝑠𝑠), of SN in terms of GN(s) and
GX(s).
(2 marks)
(e)
Hence demonstrate that SN has a Poisson distribution with mean λp.
(4 marks)
(f)
Detail and interpret how this results may be used to explore Poisson splitting.
(4 marks)
Question 3
For a certain branching process with Z0 = 1, the probability generating function for the
family size, Y, is given by
𝐺𝐺(𝑠𝑠) =
(a)
By expanding G(s), formulate
(i)
P[Y = 2]
(ii)
P[Y = 3]
1
.
(2 − 𝑠𝑠)2
(6 marks)
(b)
Compute the probability that the population ultimately becomes extinct.
(5 marks)
(c)
From the probability generating function of Y, compute the mean and variance of
the populations in the nth generation, Zn. Interpret these results, in relation to your
answer in part (b).
(9 marks)
Question 4
(a)
In the standard gambler’s ruin problem with total stake a and the gambler’s stake
k, the gambler’s probability of winning at each play is p. Apply appropriate
formulae and compute the probability of ruin and the expected duration in each
of the following cases.
(i)
a = 100, k = 5, p = 0.6
(2 marks)
(ii)
a = 80, k = 70, p = 0.45
(2 marks)
(iii)
a = 50, k = 40, p = 0.5
(2 marks)
MTH305e Copyright © 2018 Singapore University of Social Sciences (SUSS)
Examination – July Semester 2018
Page 3 of 5
(b)
Interpret and comment briefly on your answers in part (a).
(6 marks)
(c)
In a casino game based on the standard gambler’s ruin, the gambler and the dealer
each starts with 20 tokens and one token is bet on each play. The game continues
until one player has no further tokens. It is decreed that the probability that any
gambler is ruined is 0.52 to protect the casino’s profit. Solve for the probability
that the gambler wins at each play.
(8 marks)
Question 5
(a)
In a game reserve with two distinct regions, at the end of each day, animals are
observed to move between region A and region B with the following
probabilities:
•
an animal in region A in the past two days will remain in region A the next
day with probability 0.8;
•
an animal which moved from region B to region A the day before will
remain in region A the next day with probability 0.5;
•
an animal in region B in the past two days will remain in region A the next
day with probability 0.9;
•
an animal which moved from region A to region B the day before will
remain in region B the next day with probability 0.6.
We assume an animal can only be in one region in a single day.
(i)
Interpret why a 2-state model is not an appropriate Markov Chain model
for this animal movement within the game reserve.
(2 marks)
(ii)
A suitable 4-state Markov Chain model is constructed with the following
transition matrix, P.
0.8
𝑃𝑃 = � ∗
∗
∗
∗
∗
0.1
∗
∗
∗
∗ 0.5
∗
∗ �
0.6 ∗
Contrast and interpret each of the 4 states, and complete the transition
matrix (giving values to entries marked *).
(7 marks)
(iii)
By computing P2, solve for the probability that a particular animal
currently in region A, having moved from region B the day before, will
still be in region A in 2 days’ time.
(6 marks)
MTH305e Copyright © 2018 Singapore University of Social Sciences (SUSS)
Examination – July Semester 2018
Page 4 of 5
(b)
Consider the Markov Chain with states 1, 2, and 3, and transition probability
matrix, P, given by
0
𝑃𝑃 = � 1
3
0
1
2
1
2
2 �.
3
0
1 0
Compute the stationary distribution for the chain.
(5 marks)
Question 6
The partial differential equation of the probability generating function ∏(s, t) of a random
process { X(t) : t > 0 } in Lagrangian form has solutions to its auxiliary equations
and
𝑐𝑐1 = 𝜙𝜙1 (𝑠𝑠, 𝑡𝑡, ∏)
𝑐𝑐2 = 𝜙𝜙2 (𝑠𝑠, 𝑡𝑡, ∏).
(a)
Briefly describe how the two solutions to the auxiliary equations are found.
(4 marks)
(b)
Interpret and write down the form in which the general solution for ∏(s, t)
takes, in terms of c1 and c2.
(2 marks)
(c)
If X(0) = 4, formulate the initial condition and how it is used to formulate the
final solution
∏ = ∏(s, t).
(4 marks)
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MTH305e Copyright © 2018 Singapore University of Social Sciences (SUSS)
Examination – July Semester 2018
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