FACULTY OF SCIENCE
SCHOOL OF MATHEMATICS AND
STATISTICS
MATH2901
HIGHER THEORY OF
STATISTICS
Semester 1, 2016
Contents
Probability
5
Lecture Notes Part One Probability and Distribution Theory
1
1 Rand
Example 54.
Consider the statistical inferences given in the regression output for the peak
particle velocity. Conduct two hypothesis tests: one for the constant term of the regression
line, and the o
MATH2801/2901 (Higher) Theory of Statistics
Semester 1, 2009
Chapter 9 exercises solutions
1. (Step 1 in notes: state the hypotheses to be tested.)
The hypotheses to be tested are:
H0 : = 78.1 versus
MATH2801/2901 (Higher) Theory of Statistics
Semester 1, 2009
Chapter 8 exercises solutions
1. (a) Let X be a Poisson() random variable. From results for Poisson random
variables, E(X) = so the method
PROBLEM SET 7
1) Suppose that the risk-free rate is 4% and that you believe in the estimates
in the table below.
Asset
X
Y
M
E(r)
12.5%
19%
16%
30%
50%
20%
0.5
1.5
1
(a) Are assets X and Y correctly p
1
Week 10
1.1
Lecture 1
Given a random sample X1 , . . . , Xn , we have defined estimators of a parameter in a parametric
b 1 , . . . , Xn ). When we plug in the observed value
model as a function of
1
Week 9
1.1
1.1.1
Lecture 2
Statical Inference
In statistics one usually have a sequence of (random) observations (X1 , . . . , Xn ) which is call a
random sample. The aim of a statistician is often
MATH2801/2901 (Higher) Theory of Statistics
Semester 1, 2009
Chapter 4 exercises solutions
1. The transformation is dened by the relationship
y = x1/3 .
The inverse transformation is then given by
x =
MATH2801/2901 (Higher) Theory of Statistics
Semester 1, 2009
Chapter 6 exercises solutions
1. Each of the required distributions can be found using results such as that for
the sum of linear combinati
MATH2801/2901 (Higher) Theory of Statistics
Semester 1, 2009
Chapter 5 exercises solutions
1. X N (68, 162 ).
(a) P (X > 80) = P (Z > 0.75) = 0.2266
(b)
i. 18 68 = 1224
ii. 18 162 = 4608
67.88
iii. Y1
8.3 Multiple Comparisons
Anova will not be able to tell you which groups are dierent. To do this, we use multiple
comparisons - pairwire t tests. Using the Bonferonni adjusted t test we get df=n-k
xs
MATH2801/2901 Solutions to Assignment 1 Random variables
1. Moment generating functions (mgfs) contain more information about probability distributions than just the moments. For example, we can nd th
Solutions to Assignment 2 Transformations, Convergence, Inference
1. Benfords Law is named after an American engineer Frank Benford, who in 1938 set
out to nd the relative frequency of the leading dig
MATH2801/2901 (Higher) Theory of Statistics
Semester 1, 2010
Assignment 1 Random variables
Important skills to demonstrate:
Ability to use key theoretical tools to derive and explore the
properties o
MATH2801/2901 (Higher) Theory of Statistics
Semester 1, 2010
Assignment 2 Transformations, Convergence, Inference
Please include this cover sheet with your submission.
Submission date: Monday lecture,
MATH2801/2901 (Higher) Theory of Statistics
Semester 1, 2009
Chapter 2 exercises solutions
1. end line
Distribution
Discrete?
Parameters
fX (x)
px (1
Bernoulli
Bin(n, p)
Geometric
yes
yes
yes
p
n, p
p
MATH2801/2901 (Higher) Theory of Statistics
Semester 1, 2009
Chapter 1 exercises solutions
1.
fX (1) = P (X = 1) = P (winner found in rst round)
= P (dierent items in rst round)
= 1 P (same items in r
MATH2801/2901 (Higher) Theory of Statistics
Semester 1, 2009
Chapter 3 exercises solutions
1. (a) The probability function of X is
1
fX (x) =
fX,Y (x, y).
y=1
This leads to:
x
fX (x)
0
1
0.1 0.5
2
0.4
1
Week 1
1.1
1.1.1
Lecture 1
Experiments, Sample space and Events
Definition 1.1. An experiment is any process leading to recorded observations
Example 1.2. Some examples
Tossing a coin
Measuring th
1
1.1
1.1.1
Week 7
Lecture 1
Convergence of random variables - continue
Definition 1.1. A sequence of random variables (Xi )iN+ is said to converge in Lp to another
random variable X if for p 1,
lim E
7. Convergence in Probability
Lehmann 2.1; Ferguson 1
Here, we consider sequences X1 , X2 , . . . of random variables instead of real numbers. As with real numbers,
wed like to have an idea of what it
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
June 2008 Examination
MATH2901
HIGHER THEORY OF STATISTICS
(1) TIME ALLOWED - 2 hours
(2) TOTAL NUMBER OF QUESTIONS - 4
(3) ANSWE
'.
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
June 2009 Examination
MATH2901
HIGHER THEORY OF STATISTICS
(1) TIME ALLOWED - 2 hours
(2) TOTAL NUMBER OF QUESTIONS - 4
(3) AN
Sketch Solution - Assignment 1
TOTAL MARKS: 40
Question 1: Let A be a subset of
1. Let event A be independent of itself. Show that P(A) is either 1 or 0.
2. Let event A be such that P(A) = 1 or P(A)
UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
MATH2901 Higher Theory of Statistics
Assignment Two (Due on Monday 22nd May)
Names (Print):
I (We) declare that this assessment item
NAME OF CANDIDATE: . . . . . . . . . . . . . . . . . . .
STUDENT NUMBER: . . . . . . . . . . . . . . . . . . . . . .
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
June 2015 Ex
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
Semester 1 2016
MATI-I2801
Theory of Statistics
TIME ALLOWED Two (2) hours
TOTAL NUMBER OF QUESTIONS 4
ANSWER ALL QUESTIONS
THE
1.2 Lecture 2
1.2.1 Distribution arising from the normal distribution
An very important property of a family of independent normal r.vs (X 1, . . . ,Xn) is that any linear
combination of r.vs from thi
1 Week 3
1.1 Lecture 1
1.1.1 Common Discrete Distributions
Denition 1.1. A Bernoulli trial is an experiment with two possible outcomes. The outcomes are
often labelled success and failure.
A Bernoulli