Solutions to Week 3 Exercises
1.a.
f (x, y) =
1/4, if x (1, 1) and y (1, 1),
0,
otherwise.
1.b. Since the distribution is uniform, the easiest way to compute the probabilities is as the ratio of
areas
Solutions to Week 2 Exercises
1.a.
Z
M (t) =
c
exp(tx)c1 dx
0
= (ct)1 exp(tx)|c0
(ct)1 (exp(ct) 1),
=
1,
t 6= 0,
t = 0.
1.b.
Z
M (t) =
c
exp(tx)2x/c2 dx
0
=
2/(c2 )(c exp(ct)/t (exp(ct) 1)/t2 ),
t 6=
Solutions to Week 7 Exercises
1.a. The likelihood function is
n
n
Y
X
1
0 1
L(|X) =
X
exp(
Xj /)
(0 )n 0 n j=1 j
j=1
and so the log-likelihood function is
`(|X) = n0 log()
n
X
Xj / + constant.
j=1
Th
Solutions to Week 4 Exercises
1.a. and 1.b.
The joint pdf of X and Y is
fXY (x, y) =
1 r1
1 s1
x
y
exp(x)
exp(y)
(r)
(s)
for both x, y (0, ), and 0 otherwise. Since Z1 = X + Y and Z2 = X/(X + Y ), the
Solutions to Week 10 Exercises
1. The bias of T (X) is
Bias(T (X) = E(T (X) .
Let denote the estimate of using the empirical distribution Fbn . ( may differ from the original statistic
T (X), though i
Solutions to Week 5 Exercises
1.a. By the CLT,
D
n ) N (0, 2 ), which is equivalent to
n(X
D
P
n )/) N (0, 1). Since Sn ,
n(X
P
then Sn / 1. Then
n ) 1
n(X
=
Sn /
n ) D
n(X
N (0, 1)
Sn
by the Con
Solutions to Week 1 Exercises
1.a. Let cfw_Aj nj=1 be a collection of sets. Then i) (
Sn
j=1
Aj )c =
Tn
j=1
Tn
Sn
Acj and ii) ( j=1 Aj )c = j=1 Acj .
To prove these two statements, we proceed using ma
ST333/406: Exercise Sheet 4
Hand in Q1 by Thursday noon, Week 8.
1. Consider the following situation which you should model with a continuous-time
Markov process. Let (Xt )t0 denote the population siz
ST333/406: Exercise Sheet 2
Questions marked with a may not have been covered yet in lectures, so dont attempt
these yet. However, you should be able to attempt all questions before the Exercise Class
ST333/406: Exercise Sheet 1
These questions are designed to assist you in recalling techniques and results which will
be useful for the course.
Please hand in Q1 for feedback by Thursday 12 noon in We
ST333/406: Exercise Sheet 3
Some questions may not have been covered yet in lectures, so dont attempt these yet.
However, you should be able to attempt most questions before the Exercise Class in Week
ST116 Tutorial Sheet 1
Deadline: Thursday 12 October 2017, 1 pm.
This sheet has three sections, to be discussed in a meeting with your Personal Tutor in week 3.
There will be cover sheets available in
ST116 Tutorial Sheet 3
Deadline: Thursday 26 October 2017, 1 pm.
This sheet has three sections, to be discussed in a meeting with your Personal Tutor in week 3.
Only section B will be marked for credi
ST116 Tutorial Sheet 2
Deadline: Thursday 19 October 2017, 1 pm.
This sheet has three sections, and will be discussed in your assigned Tutorial Group in week 4.
Only section B will be marked for credi
DESIGNED EXPERIMENTS 2016
ST305 / ST410
Lectures 19: Response surfaces (cont.)
First-order designs
The most common first-order designs are 2k factorial designs,
PlackettBurman designs and simplex desi
DESIGNED EXPERIMENTS 2016
ST305 / ST410
Worksheet 6 (Week 7)
Exercise 1: Multiple Regression (Lecture 14)
The following data show the responses (%age of total calories obtained from complex
carbohydra
DESIGNED EXPERIMENTS 2016
ST305 / ST410
Worksheet 5 (Week 6)
Exercise 1: Random Effects (Lecture 11)
A textile company weaves a fabric on a large number of looms. It would like the looms to be
consist
DESIGNED EXPERIMENTS 2016
ST305 / ST410
Worksheet 4 (Week 5)
Exercise 1: 4 4 Lattice square (Lecture 8)
Write out a lattice square design for 16 treatments. You will need the following orthogonal
4 4
DESIGNED EXPERIMENTS 2016
ST305 / ST410
Lecture 20: Industrial Experimentation
Taguchi and Robust Parameter Design
Until quite recently statistical design methods were used in agriculture and industry
DESIGNED EXPERIMENTS 2016
ST305 / ST410
Lecture 12: Matrix approach to analysis of variance
Introduction
The basic model that we have used so far for most designs takes the form
.
Y
ij
i
j
ij
In matri
DESIGNED EXPERIMENTS 2016
ST305 / ST410
Lecture 13 Regression
Simple Linear Regression
In general we consider a single dependent variable or response Y that depends on k
independent or regressor varia
DESIGNED EXPERIMENTS 2016
ST305 / ST410
Lecture 14: Regression and matrix algebra
Preliminaries
One important aspect of experimental design, particularly in the industrial area, is to
determine the re
DESIGNED EXPERIMENTS 2016
ST305 / ST410
Lecture 15: The 2n & 3n factorial series
Introduction
Factorial experiments in which each factor operates at two levels occupy a special
place in the theory of
DESIGNED EXPERIMENTS 2016
ST305 / ST410
Lectures 18: Response surfaces
Introduction
In experimental situations we frequently use quantitative factors, and rather than
simply estimating the means it is
DESIGNED EXPERIMENTS 2016
ST305 / ST410
Lecture 11: Random effects
Introduction
Thus far we have looked at models with only one random term, the residual
or
,
ij ijk
depending on the factor structure