MAS113 Introduction to Probability and Statistics
Solutions to Exercises
1. (a) The sample space is cfw_HHH, HHT, HT H, HT T, T HH, T HT, T T H, T T T
or A A A, where A = cfw_H, T , the sample space for a single
toss. The given event is
cfw_HHT, HT H, T
MAS113 Introduction to Probability and Statistics
Supplementary Notes
Measures
In this module we are using a simplified definition of measure. The formal
definition is more complex, because it turns out that for certain sample spaces
and sensible choices
MAS113 Introduction to Probability and Statistics
1
Introduction
1.1
Studying probability theory
There are (at least) two ways to think about the study of probability theory:
1. Probability theory is a branch of Pure Mathematics. We start with some
axioms
MAS113 Introduction to Probability and Statistics
Exercises
You will be told which problems to work on each week at the lecture each Monday, and you
should try them before your class each Wednesday. Every two weeks, starting in week 3,
you will also be as
MAS113 Introduction to Probability and Statistics
Supplementary Notes
Counting
Firstly, recap that for a positive integer n,
n! := n (n 1) (n 2) . . . 2 1.
Here n! is read as n factorial. The simple observation that n! = n (n 1)!
is very useful. To be co
7
Discrete random variables
Informally, we think of a random variable as any quantity that is uncertain
to us. For example:
the number of emergency call-outs received by a fire station in a given
week;
the price of a barrel of oil in one months time;
t
MAS113 Introduction to Probability and Statistics
Proofs of theorems
Theorem 1 (De Morgans Laws). See MAS110.
Theorem 2 M1 By definition, B and A \ B are disjoint, and their union
is A. So, because m is a measure, m(A) = m(B) + m(A \ B);
rearranging gives
MAS113 Semester 1
Introduction to R
An Introduction to R
The following pages will give you a simple introduction to the software package R. As well as attempting
the various tasks, you should type in each command that is shown next to the symbol
>
(which
10
10.1
Multivariate discrete random variables
Introduction
There are various scenarios in which we may be interested in the joint behaviour of multiple random variables. For example
A share portfolio is made up of stocks in several companies listed on t
MAS113 Introduction to Probability and Statistics
Supplementary Notes
More on subjective probability
Informally, weve said that you can choose your subjective probability by
considering your fair betting odds for an event. Here, we give a formal
definiti
Running head: EFFECTIVENESS OF PRINCIPAL PREPARATION PROGRAMS
Effectiveness of Principal Preparation Programs Literature Review
Authors (Name)
College (Affiliate)
1
EFFECTIVENESS OF PRINCIPAL PREPARATION PROGRAMS
2
Effectiveness of Principal Preparation P
Data provided:
Graph Paper
AMA405
SCHOOL OF MATHEMATICS AND STATISTICS
Spring Semester 20062007
Advanced Operations Research
Answer
four
2 Hours
questions. You are advised
not
to answer more than four questions: if you
do, only your best four will be coun
AMA405
SCHOOL OF MATHEMATICS AND STATISTICS
ADVANCED OPERATIONS RESEARCH
Marks will be awarded for your best
1
(i)
four
Spring Semester 20082009
2 Hours
answers.
A company plans to build six factories on selected sites during the next
four years. Let
if f
AMA405
SCHOOL OF MATHEMATICS AND STATISTICS
Spring Semester 20072008
ADVANCED OPERATIONS RESEARCH
Answer
four
questions. You are advised
2 hours
not
to answer more than four questions: if you
do, only your best four will be counted.
1
(i)
Consider the max
PMA427
SCHOOL OF MATHEMATICS AND STATISTICS
Spring Semester 20072008
Galois Theory
Answer
four
2 hours 30 minutes
questions. If you answer more than four questions, only your best four will
be counted.
Notation: Z denotes the integers and Q the rationals.
PMA427 GALOIS THEORY EXAM SOLUTIONS 2006/07
(i) A K -automorphism of L is a function : L L such that:
is a ring homomorphism, so (0) = 0, (1) = 1, (a + b) = (a) + (b) and (ab) =
(a)(b) for all a, b L.
is a bijection.
For all a K we have (a) = a.
The Ga
SCHOOL OF MATHEMATICS AND STATISTICS
Galois Theory
Of
PMA427
University
Shefeld.
Spring Semester 20062007
2 hours 30 minutes
Answer four questions. If you answer more than four questions, only your best four will
be counted.
In this exam, as in
PMA430
SCHOOL OF MATHEMATICS AND STATISTICS
Autumn Semester
200910
Analytic Number Theory
Answer
Question 1
2 hours 30 minutes
and three other questions. You are advised
not
to answer more than
three of the questions 2 to 5: if you do, only your best thre
PMA430
SCHOOL OF MATHEMATICS AND STATISTICS
Autumn Semester 20082009
Analytic Number Theory
2 hours 30 minutes
Answer Question 1 and three other questions. You are advised not to answer more than
three of the questions 2 to 5: if you do, only your best th
PMA430
SCHOOL OF MATHEMATICS AND STATISTICS
Analytic Number Theory
Autumn Semester 2007-8
2 hours 30 minutes
Answer Question 1 and three other questions. You are advised not to answer more than
three of the questions 2 to 5: if you do, only your best thre