Student
Number
Semester 1 Assessment, 2020
School of Mathematics and Statistics
MAST20004 Probability
This exam consists of 27 pages (including this page)
Authorised materials: printed one-sided copy of the Exam or the Masked Exam made available
earlier (or an offline electronic PDF reader), two double-sided A4 handwritten or typed sheets
of notes, and blank A4 paper.
Calculators of any sort are NOT allowed.
Instructions to Students
• You should attempt all questions.
• There is a table of normal distribution probabilities and Matlab output at the end of this
question paper.
• There are 9 questions with marks as shown. The total number of marks available is 120.
Supplied by download for enrolled students only— c©University of Melbourne 2020
MAST20004 Probability Semester 1, 2020
Question 1 (10 marks)
Consider a random experiment with sample space Ω.
(a) Write down the axioms which must be satisfied by a probability mapping P defined on
the set of events of the experiment.
(b) Using the axioms, prove that for any events A and B where B ⊆ A, P(B) ≤ P(A).
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MAST20004 Probability Semester 1, 2020
(c) Let A1, A2, A3, . . . be a sequence of events in Ω. Prove that
P
( ∞⋃
n=1
An
)
≤
∞∑
n=1
P(An).
Page 3 of 27 pages
MAST20004 Probability Semester 1, 2020
Question 2 (9 marks)
A bag has three coins in it, one is fair, and the other two are weighted so the probability of a
head coming up are 512 and
1
3 , respectively. You choose a coin at random from the bag and toss
it.
(a) What is the probability of a head showing on the coin?
(b) Given a head is showing, what is the probability you tossed the fair coin?
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MAST20004 Probability Semester 1, 2020
(c) Given a head is showing, if you toss the same coin again, what is the probability that
you get a head?
Do not attempt to calculate the final answer. Leave your answer in terms of
products and quotients of fractions.
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MAST20004 Probability Semester 1, 2020
Question 3 (12 marks)
The Weibull cumulative distribution function FX is given by
FX(x) =
{
1− e−(x/β)γ , x ≥ 0
0, otherwise,
where β > 0 and γ > 0 are parameters.
(a) How could a realisation of a Weibull random variable be generated from an R(0, 1)
random number generator?
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MAST20004 Probability Semester 1, 2020
(b) Let Y = Xγ . Derive the cumulative distribution function of Y , identify the distribution
of Y , and write down E(Y ).
(c) Let Z = min(Y,M) where M is a positive finite number. Using the tail probabilities
formula for the mean, derive an expression for E(Z).
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MAST20004 Probability Semester 1, 2020
Question 4 (19 marks)
Let X and Y have joint probability density function given by
f(X,Y )(x, y) =
{
cxy, 0 < y < x < 2
0, otherwise
where c is a constant.
(a) Plot the region where f(X,Y )(x, y) is nonzero.
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MAST20004 Probability Semester 1, 2020
(b) Find the constant c.
(c) Find fY (y), the marginal probability density function of Y .
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MAST20004 Probability Semester 1, 2020
(d) Find fX|Y (x|y), the conditional probability density function of X given Y = y.
(e) Evaluate P
(
X < 32 |Y = 1
)
.
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MAST20004 Probability Semester 1, 2020
(f) Evaluate P
(
X < 32 |Y > 1
)
.
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MAST20004 Probability Semester 1, 2020
(g) Are X and Y independent? Justify your answer.
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MAST20004 Probability Semester 1, 2020
Question 5 (18 marks)
Toss a fair coin and let N be the number of tails observed before the first head appears. Now
roll a fair die N times and let X be the number of times the number “6” is observed in the N
rolls.
(a) Find E(X|N) and V (X|N).
(b) Evaluate E(X).
(c) Evaluate V (X).
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MAST20004 Probability Semester 1, 2020
(d) Find the conditional probability generating function PX|N (z).
(e) Find the probability generating function of X and identify the distribution of X by
name.
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MAST20004 Probability Semester 1, 2020
(f) Find the conditional probability mass function of N given X = x, for x = 0, 1, 2, . . ..
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MAST20004 Probability Semester 1, 2020
Question 6 (11 marks)
The price of a stock at the beginning of the trading day is 10 dollars. The price of the same stock
halfway through the trading day is S1 = 10e
X , and at the end of the trading day is S2 = 10e
Y ,
where (X,Y ) is a bivariate normal random variable with mean parameters (µX , µY ) =
(
1
2 , 1
)
,
variance parameters
(
σ2X , σ
2
Y
)
= (4, 9), and ρ = 1√
2
.
(a) Calculate the probability that the stock has a higher price halfway through the trading
day than at the beginning of the trading day.
(b) If you buy one share of the stock at the start of the trading day and sell it halfway
through the trading day, how much money would you expect to make (losses count as
negative)?
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MAST20004 Probability Semester 1, 2020
(c) Given the price of the stock at the end of a trading day was 10e4 dollars, what is the
probability the price halfway through that day was lower than 10 dollars?
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MAST20004 Probability Semester 1, 2020
Question 7 (20 marks)
Let X
d
= R(0, 1).
(a) For t 6= 0, derive the moment generating function of X.
(b) State the value of MX(0) justifying your answer.
(c) Derive the moment generating function of Y = X − 12 .
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MAST20004 Probability Semester 1, 2020
(d) Calculate the skewness of X, that is, κ3 = E
(
(X − µX)3
)
, and give an interpretation
for your answer.
(e) Calculate the kurtosis of X, that is, κ4 = E
(
(X − µX)4
)
− 3σ4.
Is the kurtosis of X positive, negative, or zero? Justify your answer and give an inter-
pretation for it.
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MAST20004 Probability Semester 1, 2020
(f) Let X1, X2, . . . be a sequence of independent random variables where each one has the
same distribution as X
d
= R(0, 1). Derive the moment generating function of
Yn =
X1 +X2 + . . .+Xn − n
2√
n
.
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MAST20004 Probability Semester 1, 2020
(g) Use the moment generating function of Yn to prove that Yn converges to N
(
0, 112
)
in
distribution as n→∞.
Hint: you may wish to use ex ≈ 1 + x+ x22 + x
3
6 for small x.
Page 21 of 27 pages
MAST20004 Probability Semester 1, 2020
Question 8 (10 marks)
Let X
d
= exp(3) and W
d
= N(−6, 25).
(a) Explain how a realisation of U
d
= R(0, 1) can be used to generate a realisation of X.
(b) Explain how a realisation of Z
d
= N(0, 1) can be used to generate a realisation of W .
(c) Explain how to calculate an estimate of E(XW ) from 1,000 realisations of the random
variable U
d
= R(0, 1) and 1,000 realisations of the random variable Z
d
= N(0, 1).
Note that the “rand” command in Matlab generates a realisation of U and the “randn”
command generates a realisation of Z.
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MAST20004 Probability Semester 1, 2020
(d) Let Y =
∫ X
0
3e−3tdt. What is the distribution of Y ? Justify your answer.
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MAST20004 Probability Semester 1, 2020
Question 9 (11 marks)
The weather in Melbourne on any day can be in one of three states: 1=“wet”, 2=“cold”, or
3=“miserable”. The state of the weather can be described by a Markov chain model with
transition probability matrix
P =

3
10
1
4
9
20
2
5
1
10
1
2
1
4
2
5
7
20
 .
(a) Which underlying assumption has been made so that this situation can be modelled as
a Markov chain?
(b) For this part justify your answers where necessary, and you may wish to use the Matlab
output at the end of this question paper to reduce your calculations.
Suppose on Monday, the weather is cold.
What is the probability that the weather will be
(i) miserable on Wednesday of the same week?
(ii) wet on Thursday of the same week?
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MAST20004 Probability Semester 1, 2020
(iii) cold on all days from Tuesday to Thursday of the same week?
(c) Explain two methods that can be used for finding the stationary distribution of the
Markov chain, one of which uses the Matlab output at the end of this question paper.
Give the stationary distribution of the Markov chain and interpret your answer.
End of Exam—Total Available Marks = 120
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MAST20004 Probability Semester 1, 2020
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MAST20004 Probability Semester 1, 2020
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