© 2021 Imperial College London Page 1

MATH50010

BSc, MSci and MSc EXAMINATIONS (MATHEMATICS)

May-June 2021

This paper is also taken for the relevant examination for the

Associateship of the Royal College of Science

Probability for Statistics

SUBMIT YOUR ANSWERS AS SEPARATE PDFs TO THE RELEVANT DROPBOXES ON

BLACKBOARD INCLUDING A COMPLETED COVERSHEET WITH YOUR CID NUMBER,

QUESTION NUMBERS ANSWERED AND PAGE NUMBERS PER QUESTION.

Date: Tuesday, 18 May 2021

Time: 09:00 to 11:00

Time Allowed: 2 hours

Upload Time Allowed: 30 minutes

This paper has 4 Questions.

Candidates should start their solutions to each question on a new sheet of paper.

Each sheet of paper should have your CID, Question Number and Page Number on the

top.

Only use 1 side of the paper.

Allow margins for marking.

Any required additional material(s) will be provided.

Credit will be given for all questions attempted.

Each question carries equal weight.

1. (a) Suppose (Ω, F, Pr) is a probability space. Determine whether or not each of the following

statements is true or false in general. Give proofs or counterexamples as appropriate.

(i) F is closed under symmetric differences: if A ∈ F and B ∈ F, then A4B ∈ F, where

A4B = (A ∪ B)\(A ∩ B).

(2 marks)

(ii) If A1, A2, . . . An ∈ F then

Pr \n

i=1

Ai

!

≥

Xn

i=1

Pr(Ai) − (n − 1).

(4 marks)

(iii) If A ∈ F is such that Pr(A) = 1, then A = Ω.

(2 marks)

(iv) F is closed under arbitrary unions: if Ai ∈ F for all i ∈ I, then S

i∈I Ai ∈ F.

(3 marks)

(v) If A1, A2, . . . is a sequence of events such that Pr(An) = 0 for all n ≥ 1, then

Pr S

n≥1 An

= 0.

(3 marks)

(b) Suppose a coin is flipped twice, so that the sample space for the experiment is Ω =

{HH, HT, T H, T T}. Let X, Y : Ω → R be defined as follows

X(ω) =

1 ω ∈ {HH, HT}

0 ω ∈ {T H, T T}

Y (ω) =

1 ω ∈ {HH, T T}

0 ω ∈ {HT, TH}

(i) State whether or not each of the functions X and Y is a random variable with respect

to the algebra

F = {∅, {HH, HT}, {T H, T T}, Ω}

of subsets of Ω. Justify your answers.

(4 marks)

(ii) Determine FZ, the smallest algebra of subsets of Ω with respect to which Z = XY is a

random variable.

(2 marks)

(Total: 20 marks)

MATH50010 Probability for Statistics (2021) Page 2

2. The random variable X has probability density function given by

fX(x|θ) = θ(1 − x)

θ−1

0 < x < 1,

and zero otherwise, where θ > 0.

(a) Find the cumulative distribution function FX(·|θ).

(2 marks)

(b) With reference to FX, explain why X is a continuous random variable.

(1 mark)

(c) Give examples of functions g : R → R such that Y = g(X) is i) a discrete random variable

ii) a random variable that is neither discrete nor continuous. Justify your answers.

(4 marks)

(d) Given a random sample U1, U2, . . . Un ∼ Uniform(0, 1), explain how to generate a random

sample of size n from the distribution of X. State clearly any results that you use.

(4 marks)

(e) Show that the random variable Xn = min{U1, . . . , Un} has the same distribution as X for a

particular value of θ, which you should specify.

(4 marks)

(f) Show that Xn

P−→ 0 as n → ∞.

(2 marks).

(g) Find a deterministic (i.e. non-random) sequence (an)n≥1 such that anXn

D

−→ Z, where Z is

a non-degenerate random variable, whose distribution you should specify.

(3 marks)

(Total: 20 marks)

MATH50010 Probability for Statistics (2021) Page 3

3. (a) Consider the standard bivariate Normal random variable Z = (X, Y ) with probability density

function

fXY (x, y) = 1

2π

exp

−

1

2

x

2 + y

2

, (x, y) ∈ R2

.

(i) Determine the joint distribution of (S, T) =

1

2

(X1 + X2),

1

2

(X1 − X2)

. State whether

or not S and T are independent, with brief justification. (4 marks)

(ii) Find the joint distribution of (R, Θ) where X = R cos Θ and Y = R sin Θ. State

whether or not R and Θ are independent . (4 marks)

(iii) Find the distribution of V =

Y

X

. (3 marks)

(b) Suppose the random variable Z has E(Z) = µ and Var(Z) = σ

2

.

(i) Show that for all α > 0,

Pr(Z − µ ≥ α) ≤ Pr

(Z − µ + y)

2 ≥ (α + y)

2

for all y > 0.

(2 marks)

(ii) Deduce that for any α > 0,

Pr(Z − µ ≥ α) ≤

σ

2

σ

2 + α2

.

(3 marks)

(iii) Suppose now that µ = 0. For given values of σ

2

and α, show that the upper bound

for Pr(Z ≥ α) established in (ii) is the sharpest possible, by constructing an example in

which equality holds.

(4 marks)

(Total: 20 marks)

MATH50010 Probability for Statistics (2021) Page 4

4. The sequence (Xn)n≥0 is a Markov chain with transition matrix

0

1

3

2

3

0 0 0

0

1

4

3

4

0 0 0

0

1

2

1

2

0 0 0

0

1

3

1

3

1

3

0 0

0

1

2

0

1

2

0 0

0 0 1

2

0

1

2

0

(a) Draw the transition diagram for (Xn).

(3 marks)

(b) Determine the communicating classes of the chain, stating whether each class is recurrent or

transient.

(3 marks)

(c) Determine the period of each communicating class. (3 marks)

(d) Suppose that the initial distribution of the chain is uniform on the set {1, 2, 5, 6}. Find

Pr(X1 = 2).

(3 marks)

(e) If the initial distribution of the chain is π0 = (0,

1

2

,

1

2

, 0, 0, 0), determine the limiting probability

distribution of the chain.

(3 marks)

(f) Consider now the Markov chain Yn on the state space {1, 2}, with transition matrix

1 − α α

β 1 − β

,

where 0 < α, β < 1.

Write down the log likelihood of the parameters α and β given a realization (y0, y1, y2, . . . , yn)

of the chain, where each yk ∈ {1, 2}.

Derive the maximum likelihood estimators of α and β. Give your answer in terms of nij , the

number of jumps from state i to state j for i, j ∈ {1, 2}. Assume each nij > 0.

(5 marks)

(Total: 20 marks)

MATH50010 Probability for Statistics (2021) Page 5