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— 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

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