MAST20004 Probability 2014
Tutorial Set 7
1. Let X be a random variable with distribution function FX (x) and Y = (X) where is a strictlydecreasing function mapping SX to some set SY .
(a) Derive an expression for FY (y) in terms of FX (x), that is valid

MAST20004 Probability - 2014
Assignment 4 - Solutions
1. (a) The MGF of X 2 is given by
2
M (t) = E[etX ] =
1
2
2
etx ex
2 /(2 2 )
dx =
1
2
2
e
x
2
1
2t
2
dx.
This integral will converge as long as
t<
1
2 2
in which case it is equal to
M (t) = 1 2 2 t
1

MAST20004 Probability 2014
Tutorial Set 4
1. The data below are taken from a study by Geissler and are given in Fishers book Statistical methods
for research workers. A large number (53,680) of German families with 8 children were contacted, and
for each

MAST20004 Probability 2014
Tutorial Set 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 success

MAST20004 Probability 2014
Tutorial Set 5
1. A Poisson distribution has P (X = 1) = P (X = 2). Find P (X = 0).
Solution: For a Poisson random variable with parameter ,
k e
.
k!
P (X = k) =
So P (X = 1) = P (X = 2) implies that
2 e
2
2 which, in turn impli

MAST20004 Probability 2014
Tutorial Set 1
1. The Prisoners Paradox relates to a situation that is similar to the Monty Hall game.
It can be described at as follows:
Three prisoners (A, B and C) have been sentenced to death. However, the governor has rando

MAST20004 Probability 2014
Tutorial Set 6
1. If X is the number of heads obtained in 100 tosses of a fair coin, use an appropriate approximation to
calculate P (X > 65) and P (X 40).
Solution: X is distributed binomially with parameters n = 100 and p = 1/

MAST20004 Probability 2014
Tutorial Set 3
Tutorial problems:
1. The newsvendor 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 from their supp

MAST20004 Probability 2014
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

MAST20004 Probability 2014
Tutorial Set 10
d
1. If X = R(0, ) and Z = sin X , nd V (Z) and compare this with the approximate value calculated
2
using V (X) ()2 V (X).
Solution:
2
fX (x) =
0
=
=
=
=
otherwise.
2 /2
sin(x)dx
0
2
/2
[ cos(x)]0
2
.
E(Z) =
E

MAST20004 Probability 2014
Tutorial Set 8
1. Each day a grocer makes $X from sales and incurs overhead costs $Y . Assume that X and Y have
2
2
a bivariate normal distribution with parameters X = 1000, Y = 50, X = 10000, Y = 25 and
= 0.7.
(a) What is the

MAST20006 Probability for Statistics /MAST90057 Elements of Probability
Assignment 1, Semester 1 2016
Due date: 4pm, Friday March 18.
Your assignment should show all working and reasoning. Marks will be given for
method as well as for correct answers.
A