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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— cUniversity of Melbourne 2020
MAST20004 Probability Semester 1, 2020
Question 1 (10 marks)
Consider a random experiment with sample space.
(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).
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?
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.
Page 5 of 27 pages
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?
Page 6 of 27 pages
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).
Page 7 of 27 pages
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.
Page 8 of 27 pages
MAST20004 Probability Semester 1, 2020
(b) Find the constant c.
(c) Find fY (y), the marginal probability density function of Y .
Page 9 of 27 pages
MAST20004 Probability Semester 1, 2020
(d) Find fX|Y (x|y), the conditional probability density function of X given Y = y.
(e) Evaluate P