Academic excellence for business
and the professions
Bayes Business School
MSc Actuarial Science
Alternative Timed Assessment
Module Code
SMM047
Module Title
Probability & Mathematical Statistics (For subject CS1 of the Institute and Faculty of Actuaries Examinations)
January 2022 3hrs – 45mins
Division of Marks:
The number of marks allocated is shown at the end of each question.
Where marks have been quoted for parts of questions, these are intended
to be a helpful guide to candidates.
Instructions to students:
Candidates should answer ALL of the questions.
Candidates should begin each question on a new page.
Your work should be in your own words, it should NOT contain material copied straight from
lecture notes, textbooks, or other resources.
If you do rely on external sources within your answer, these should be properly
referenced/cited.
You are expected to show all necessary working to obtain your final solution.
If this is not done, then marks will be deducted even when the correct numerical solution is
obtained.
This paper contains NINE questions and comprises SEVEN pages including the
title page
Internal Examiner: Dr. Russell Gerrard
External Examiner: Professor. Goran Peskir
QUESTION 1
In 2019 15,000 schoolchildren sat a state Mathematics examination. The average mark was 63.7,
standard deviation 14.4. The examiners’ intention when setting the paper was that the mean mark
should be 65, standard deviation 15.
(i) Test whether there is evidence at the 5% level of significance that the examination was more
difficult than the examiners intended. [2 marks]
(ii) The examiners decide to transform the marks in such a way that the transformed sample has
sample mean 65, sample standard deviation 15.
(a) Calculate the parameters of an appropriate linear transformation, Yi = a+bXi
. [2 marks]
(b) The sample coefficient of skewness of the original marks is ?0.03. What is the sample
coefficient of skewness of the transformed data? Give a reason for your answer. [1 mark]
[Total: 5 marks]
QUESTION 2
A discrete random X has probability generating function
GX(t) = 1
2
(1 + t)e
λ(t?1)
(i) Calculate the expectation of X and the probability that X takes the value 0. [4 marks]
(ii) Identify the distributions of two discrete random variables U and V whose probability generating functions are
GU (t) = 1
2
(1 + t), GV (t) = e
λ(t?1)
.
[2 marks]
(iii) Is it possible to deduce that, if U and V are independent, their sum U + V has the same
distribution as X? Give a reason for your answer. [1 mark]
(iv) Write down E[U], E[V ], P[U = 0] and P[V = 0] and use these to verify your answer to (i).
[3 marks]
[Total: 10 marks]
Page 2 of 7
QUESTION 3
A discrete random variable X has probability function
pX(x) = 1
? log(1 ? θ)
·
θ
x
x
, x = 1, 2, . . .
(i) Find an expression for the expectation of X as a function of θ. [3 marks]
(ii) A researcher announces that, based on a set of 50 observations x1, . . . x50 of the variable X,
she has calculated the method of moments estimate of θ to be 0.6. Assuming she is correct,
what is the value of the sample mean x? [2 marks]
(iii) Show that the maximum likelihood estimator for the parameter θ gives the same value as the
method of moments estimator. [4 marks]
[Total: 9 marks]
QUESTION 4
A collection X1, . . . , Xn of observations are taken from a Normal distribution with unknown mean
μ but with variance known to be equal to 1.
Knowing that the quantity being modelled is positive, an analyst applies as the prior distribution
for μ an exponential distribution with rate parameter λ.
(i) Show that the posterior density of μ given the observations satisfies
π(μ; x) ∝
(
exp
?
n
2
