ST102 Outline solutions to Exercise 12
1. (a) We have:
E
k
X
!
ai Xi
=
k
X
i=1
E(ai Xi ) =
k
X
i=1
ai E(Xi ).
i=1
(b) We have, noting that Xi and Xj are independent for i 6= j and hence uncorrelated,
i.e. with a zero covariance:
!
!2
k
k
k
X
X
X
Var
= E

ST102 Outline solutions to Exercise 7
1. We have:
MX (t) = E(etX ) =
k
X
x=1
=
=
etx
1
k
et e2t
ekt
+
+ +
k
k
k
1 t
(e + e2t + + ekt ).
k
The bracketed part of this expression is a geometric series with a first term of et and a
common ratio also of et . H

ST102 Exercise 12
In this exercise you will practise deriving expressions for the mean and variance of a linear
combination of random variables in Question 1. Question 2 requires you to explore the
properties of an estimator for the parameter of a Bernoul

ST102 Outline solutions to Exercise 20
1.* (a) Approach 1: We minimise:
L() =
X
2i =
i
X
(yi xi )2 .
i
Differentiating:
X
dL()
= 2
xi (yi xi ).
d
i
b to obtain:
We now set the first derivative equal to zero and solve for the LSE, ,
P
x i yi
b = Pi 2 .
i x

ST102 Outline solutions to Exercise 13
as the estimator of the population mean, which is the
1. (a) We use the sample mean, X,
method of moments estimator (MME), the least sqaures estimator (LSE), and also
the maximum likelihood estimator (MLE). For the

ST102 Outline solutions to Exercise 18
1.* We need to calculate the following:
3
X
A = 1
XAi ,
X
3
i=1
4
5
X
B = 1
X
XBi ,
4
i=1
and:
3
P
=
X
XAi +
4
P
X
D = 1
X
XDi
3
i=1
XBi +
5
P
XCi +
i=1
i=1
i=1
3
X
C = 1
X
XCi ,
5
3
P
i=1
XDi
i=1
15
.
Alternatively:

Chapter 7
Point estimation
7.1
Synopsis of chapter
This chapter covers point estimation. Specifically, the properties of estimators are
considered and the attributes of a desirable estimator are discussed. Techniques for
deriving estimators are introduced

ST102 Outline solutions to Exercise 14
1. (a) We have x
= 2.115, and the top 0.5th percentile of N (0, 1) is z0.005 = 2.576. Hence
a 99% confidence interval for is:
x
z0.005 / n = 2.115 2.576 1.1/ 11 = (1.260, 2.969).
(b) The sample median is 1.79, and

ST102 Outline solutions to Exercise 4
1. Let N denote a normal environment, S denote a severe environment and F denote a
failure. We have P (N ) = 0.95, P (S) = 0.05, P (F | N ) = 0.02 and P (F | S) = 0.5. The
total probability formula then gives:
P (F )

ST102 Exercise 10 (2014-15)
In this exercise you will practise deriving and working with sampling distributions of statistics.
to approximate the
Question 2 concerns, in particular, the use of the central limit theorem
of the 12 and
definitions
the
4
conc

ST102 Exercise 19
In this exercise you will practise fundamental aspects of the simple linear regression model.
Question 1 considers properties of the residuals. Question 2 relaxes the assumption that the
xi s are constants, and instead treats X as a rand

Chapter 4
Common distributions of random
variables
4.1
Synopsis of chapter content
This chapter formally introduces common families of probability distributions which
can be used to model various real-world phenomena.
4.2
Learning outcomes
After completin

ST102 Outline solutions to Exercise 8
1. Using Table 3 of Murdoch and Barnes Statistical Tables we have the following
probabilities.
(a) P (0 < Z < 1.2) = P (Z > 0) P (Z > 1.2) = 0.5 0.1151 = 0.3849.
(b) P (0.68 < Z < 0) = P (Z < 0) P (Z < 0.68) = 0.5 P (

Chapter 11
Linear regression
11.1
Synopsis of chapter
This chapter covers linear regression whereby the variation in a continuous dependent
variable is modelled as being explained by one or more continuous independent
variables.
11.2
Learning outcomes
Aft

ST102 Exercise 14
In this exercise you will practise calculating confidence intervals. Question 1 assumes the
population standard deviation is known and also involves sample size determination.
Question 2 uses the sample standard deviation. In Question 3,

Chapter 2
Probability theory
2.1
Synopsis of chapter
Probability is very important for statistics because it provides the rules which allow us
to reason about uncertainty and randomness, which is the basis of statistics.
Independence and conditional proba

Chapter 6
Sampling distributions of statistics
6.1
Synopsis of chapter
This chapter considers the idea of sampling and the concept of a sampling distribution
for a statistic (such as a sample mean) which must be understood by all users of
statistics.
6.2

Chapter 9
Hypothesis testing
9.1
Synopsis of chapter
This chapter discusses hypothesis testing which is used to answer questions about an
unknown parameter. We consider how to perform an appropriate hypothesis test for a
given problem, determine error pro

ST102 Exercise 1, Outline solutions
1. (a) Insert your guess here. Chances are that you got the average correct, without
even thinking that it could possibly be anything else. The standard deviation was
probably less obvious.
(b)
i. Using the rules of sum

ST102 Outline solutions to Exercise 11
1. Each possible sample has an equal probability of occurrence of 1/6.
is:
distribution of X
Possible samples
(1, 2)
(2, 1)
(1, 3)
(3, 1)
(2, 3)
(3, 2)
=x
Sample mean, X
1.5
2
2.5
=x
Relative frequency, P (X
)
1/3

Week 11 Quiz answers
Question 1
Here are two Lorenz Curves. Which of them represents the higher level of
inequality?
a. Can't Say for Sure
b. Lorenz Curve B
c. Lorenz Curve A
Question 2
If Sweden has a lower Gini coefficient than the UK, does Sweden have

ST102 Exercise
In
B
exercise you will practise working with some common probability distributions.
Questi s 1,2 and 3 involve the normal distribution. Question 4 concerns the exponential
distri
ion' Questions 5 and 6 aiso involve the normal clistribution,

ST102 Exercise 7
In this exercise you will practise working witir some common discrete distributions - the discrete
uniform distribution (Question 1), the binomial distribution (Questions 2,3 and 4), including
the Poisson approximation of the binomial (Qu

ST102 Exercise
In
9
cise you will practise various topics related to multivariate random variables.
1 and 2 cover discrete bivariate distributions. Questions 3, 4 and 5 concern
inations of independent random variables. Question 6 is a theoretical exercise

ST102 Exercise 5
In this exercise you will
practise conditional probability (Question 1), aspects of discrete
random variables including deriving probability functions (Questions 2 and 3), expected values
(Question 4), the moment generating function (Ques

ST102 Exercise 1 (2014-15)
Practicalities: Your
class teacher
will explain to you the dates when solutions to future
will be marked and returned to you, during your
first class. You should spend a reasonable amount of time on each set of exercises. Do not

ST102 Exercise
6
you will practise the key elements of distributions of continuous random
(probability density functions and cumulative distribution functions) and use them
to deri probabilities) means, variances and medians for such distributions. Each o

ST102 Exercise 3 (2014-15)
In this exercise you will practise the calculation of classical probabilities (Question 1), the
calculation of probabilities for independent events (Question 2), the definition of independent
events (Question 3), the total proba

ST102 Exercise 2
In this exercise you will practise definitions and operations on sets (Questions 1 to 3), counting
4) and performing some logic analysis using sets and a Venn diagram
(Question 5).
outconpes (Question
1. Ilet A, B, and C be events in a sa

ST102 Exercise 4 (20I4-I5)
In this exercise you will practise the total probability formula and Bayes' theorem (Questions
1, 2 and 3), the calculation of probabilities for independent events (Questions 4 and 5) and,
for discrete random variables, the defi