7

.r'
MAST20004 Probability
Tutorial Set

2014
3
T\rtorial problems:
1. The newsuend,or problem is faced by retailers who sell perishable goods. (Newspapers could be the
ultimate perishable good.) Newsvendors have to decide how many papers to order fro
MAST20004 Probability 2016
Assignment 3
If you didnt already hand in a completed and signed Plagiarism Declaration Form
(available from the LMS or the M&S webpage), please do so and attach it to the front
of this assignment.
Assignment boxes are located o
MAST20004 Probability
Tutorial Set 11
1. Let X and Y be independent random variables, with known moment generating functions MX (t) and
MY (t) and Z be such that P (Z = 1) = 1 P (Z = 0) = p (0, 1). Compute the moment generating
function of the random vari
MAST20004 Probability
Tutorial Set 10
d
1. If X = R(0, 2 ) and Z = sin X , find V (Z) and compare this with the approximate value calculated
using V (X) 0 ()2 V (X).
2. Let N 0 be an integervalued random variable with E(N ) = a, V (N ) = b2 and X1 , X2 ,
MAST20004 Probability 2016
Assignment 4
If you didnt already hand in a completed and signed Plagiarism Declaration Form
(available from the LMS or the departments webpage), please do so and attach it to
the front of this assignment.
Assignment boxes are l
Administration
Web site: accessible from through the Learning Management System
Lectures
Consultation hours
Tutorial/Computer Lab classes
Homework
Assessment
Slide 1
Probability is interesting
The Monty Hall Game
The Birthday Problem
The BusStop
MAST20004 Probability  2015
Assignment 4  Solutions
1. (a) If Xs is standardised X, P (100eX > 100) = P (X > 0) = P (Xs > 2) .98, since
Xs is standard normal.
(b) On average you would make
E[100eX ] 100 = 100(e2 E[eXs ] 1) = 100(e5/2 1).
p
(c) The corre
MAST20004 Probability 2014
Assignment 1
Please complete and sign the Plagiarism Declaration Form (available from the LMS
or the departments webpage), which covers all work submitted in this subject. The
declaration should be attached to the front of your
MAST20004 Probability  2014
Assignment 1  Solutions
1. Let A, B, and C be the events that the part came from the respective machine and D
be the event the part is defective. The problem says that
P (A) = .35,
P (B) = .4,
P (DA) = .05,
P (C) = .25,
P (D
MAST20004 Probability 2015
Assignment 4
If you didnt already hand in a completed and signed Plagiarism Declaration Form
(available from the LMS or the M&S webpage), please do so and attach it to the front
of this assignment.
Assignment boxes are located o
MAST20004 Probability  2014
Assignment 2  Solutions
1. (a) Since densities integrate to one and X is a random variable between 0 and 1:
Z 1
1
Ca
1
a
x (1 x)dx = Ca
1 = Ca
=
.
a+1 a+2
(a + 1)(a + 2)
0
So Ca = (a + 1)(a + 2).
(b) The CDF of X, FX (x) equa
MAST20004 Probability  2014
Assignment 3  Solutions
1. By the definition of conditional probability the joint pmt of (X, Y ) is for x = 1/2, 1/3, 1/4
and k = 0, 1, . . .
1
f (x, k) = (1 x)k x.
3
(a) The law of total probability implies the marginal pmt
MAST20004 Probability
Assignment 1
Please complete and sign the Plagiarism Declaration Form (available from the LMS
or the departments webpage), which covers all work submitted in this subject. The
declaration should be attached to the front of your first
MAST20004 Probability 2016
Assignment 2
If you didnt already hand in a completed and signed Plagiarism Declaration Form
(available from the LMS or the departments webpage), please do so and attach it to
the front of this assignment.
Assignment boxes are l
MAST20004 Probability
Assignment

2OL4
1
Please complete and sign the Plagiarism Declaration Form (available from the LMS
or the department's webpage), which covers all work submitted in this subject. The
declaration should be attached to the front of yo
rl
I
MAST20004 ProbabilitY

2OL4
Tutorial Set 1
l'ThePrisoner,sParadoxrelatestoasituationthatissimilartotheMontyHallgame.
It
can be described at as follows:
Threeprt'so,ners(A,Band'C)hauebeensentenced'tod,eath,Howeuer,thegouernorlt,asrandomly
h'e aslcs t
,/
MAST20004 Probability
Tutorial Set

2OL4
2
Tutorial problems:
1. In lectures, we discussed the experiment where we repeatedly roll a die until we get a six. In this
question we think about this experiment in more detail. Assume that the outcomes of su
MAST20004 Probability
2OL4

Tutorial Set 5
1. A Poisson distribution has P(X
2.
: I) : P(X :
2). Find P(X
:
0).
be the number of zeros in n :50 independent random decimal digits (each digit from 0, 1, . . . ,9
has equal probability of occurring). Find
Le

62O2OL
Probability
Computer Lab
2O14
3
This lab considers various aspects of the random experiment of throwing two fair dice' In particular we
o display the partitions of the sample
space 'generated' by different random variables;
o calculate the theo
/
62020l Probability
Computer Lab
2OI4
2
The aim of this lab is to
.
use MATLAB to simulate a simple die experiment to help test whether two events
positively or negatively related to each other;
o
use
use
B
MATLAB to investigate the Multipie Choice Exa

/
62A2OL
ProbabilitY
Computer Lab
2OL4
1
The aim of this lab is to
r
recail some basic features of MATLAB;
o
use
MATLAB to simulate a simpie die experiment;
o
use
MATLAB to investigate the bus stop paradox from lectures.
you are unlikely to have much
62O2OL
Probability
Computer Lab
2O1'4
5
In this lab you
o investigate the shape of the pmf for a negative binomial random variable for different values of its
parameters.
r
study the spatial distribution of various facilities in the city of Coventry in
The University of Melbourne
Department of Mathematics and Statistics
620201 Probability
Semester 1 Exam June 24, 2009
Exam Duration: 3 Hours
Reading Time: 15 Minutes
This paper has 5 pages
Authorised materials:
Students may bring one doublesided A4 shee
MAST20004 Probability
Tutorial Set 9
1. Let X and Y be independent rvs with E (X) = E (Y ) = 5, V(X) = 1 and V(Y ) = 2 > 1. Put
Z = aX + (1 a)Y , 0 a 1. Find (i) the value of a that minimizes V(Z) and that minimum value,
and (ii) the value of a that maxim
MAST20004 Probability  2014
Assignment 4  Solutions
1. (a) The MGF of X 2 is given by
Z
Z
2
1
1
1
2
2
2
2
x
2t
etx ex /(2 ) dx =
e 2 2
dx.
M (t) = E[etX ] =
2
2
This integral will converge as long as
t<
1
2 2
in which case it is equal to
1/2
M (t)
MAST20004 Probability 2014
Assignment 4
If you didnt already hand in a completed and signed Plagiarism Declaration Form
(available from the LMS or the departments webpage), please do so and attach it to
the front of this assignment.
Assignment boxes are l
MAST20004 Probability 2014
Assignment 2
If you didnt already hand in a completed and signed Plagiarism Declaration Form
(available from the LMS or the departments webpage), please do so and attach it to
the front of this assignment.
Assignment boxes are l
The University of Melbourne
Department of Mathematics and Statistics
620—20]; Probability
Semester 1 Exam — June 249 2010
Exam Duration: 3 Hours
Reading Time: 15 Minutes
This paper has 5 pages
Authorised materials:
Students may bring one double—side
The University of Melbourne
Department of Mathematics and Statistics
620—201 Probability
Semester 1 Exam —— June 24, 2009
Exam Duration: 3 Hours
Reading Time: 15 Minutes
This paper has 5 pages
Authorised materials:
Students may bring one double—sided A4