MATH2010-E1

The University of Nottingham

SCHOOL OF MATHEMATICAL SCIENCES

A LEVEL 2 MODULE, SPRING SEMESTER 2019-2020

PROBABILITY MODELS AND METHODS

Suggested time to complete: TWO Hours THIRTY Minutes

Paper set: 18/05/2020 - 10:00

Paper due: 26/05/2020 - 10:00

Answer all SIX questions

Your solutions should be written on white paper using dark ink (not pencil), on a tablet, or

typeset. Do not write close to the margins. Your solutions should include complete

explanations and all intermediate derivations. Your solutions should be based on the material

covered in the module and its prerequisites only. Any notation used should be consistent with

that in the Lecture Notes.

Guidance on the Alternative Assessment Arrangements can be found on the Faculty of Science

Moodle page: https://moodle.nottingham.ac.uk/course/view.php?id=99154#section-2

Submit your answers as a single PDF with each page in the correct orientation, to the

appropriate dropbox on the module’s Moodle page. Use the standard naming

convention for your document: [StudentID]_[ModuleCode].pdf. Please check the

box indicated on Moodle to confirm that you have read and understood the statement

on academic integrity: https://moodle.nottingham.ac.uk/pluginfile.php/6288943/mod_

tabbedcontent/tabcontent/8496/FoS%20Statement%20on%20Academic%20Integrity.pdf

A scan of handwritten notes is completely acceptable. Make sure your PDF is easily readable

and does not require magnification. Text which is not in focus or is not legible for any other

reason will be ignored. If your scan is larger than 20Mb, please see if it can easily be reduced

in size (e.g. scan in black & white, use a lower dpi — but not so low that readability is

compromised).

Staff are not permitted to answer assessment or teaching queries during the assessment

period. If you spot what you think may be an error on the exam paper, note this in your

submission but answer the question as written. Where necessary, minor clarifications or

general guidance may be posted on Moodle for all students to access.

Students with approved accommodations are permitted an extension of 3 days.

The standard University of Nottingham penalty of 5% deduction per working day will

apply to any late submission.

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MATH2010-E1

Academic Integrity in Alternative Assessments

The alternative assessment tasks for summer 2020 are to replace exams that would have

assessed your individual performance. You will work remotely on your alternative assessment

tasks and they will all be undertaken in “open book” conditions. Work submitted for

assessment should be entirely your own work. You must not collude with others or employ the

services of others to work on your assessment. As with all assessments, you also need to avoid

plagiarism. Plagiarism, collusion and false authorship are all examples of academic misconduct.

They are defined in the University Academic Misconduct Policy at: https://www.nottingham.ac.

uk/academicservices/qualitymanual/assessmentandawards/academic-misconduct.aspx

Plagiarism: representing another person’s work or ideas as your own. You could do this by

failing to correctly acknowledge others’ ideas and work as sources of information in an

assignment or neglecting to use quotation marks. This also applies to the use of graphical

material, calculations etc. in that plagiarism is not limited to text-based sources. There is

further guidance about avoiding plagiarism on the University of Nottingham website.

False Authorship: where you are not the author of the work you submit. This may include

submitting the work of another student or submitting work that has been produced (in whole

or in part) by a third party such as through an essay mill website. As it is the authorship of an

assignment that is contested, there is no requirement to prove that the assignment has been

purchased for this to be classed as false authorship.

Collusion: cooperation in order to gain an unpermitted advantage. This may occur where you

have consciously collaborated on a piece of work, in part or whole, and passed it off as your

own individual effort or where you authorise another student to use your work, in part or

whole, and to submit it as their own. Note that working with one or more other students to

plan your assignment would be classed as collusion, even if you go on to complete your

assignment independently after this preparatory work. Allowing someone else to copy your

work and submit it as their own is also a form of collusion.

Statement of Academic Integrity

By submitting a piece of work for assessment you are agreeing to the following statements:

1. I confirm that I have read and understood the definitions of plagiarism, false authorship

and collusion.

2. I confirm that this assessment is my own work and is not copied from any other person’s

work (published or unpublished).

3. I confirm that I have not worked with others to complete this work.

4. I understand that plagiarism, false authorship, and collusion are academic offences and I

may be referred to the Academic Misconduct Committee if plagiarism, false authorship or

collusion is suspected.

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1 MATH2010-E1

SECTION A

1. (a) Suppose that and have joint probability density function

,(, ) = {

6, > 0, > 0, + < 1,

0, otherwise.

From this it follows that the marginal density functions are

() = {

6(1 − ), ∈ (0, 1),

0, otherwise

and () = {

3(1 − )2, ∈ (0, 1),

0, otherwise.

Determine the probability density function of = + . [8 marks]

(b) i) Find the distribution of = + , where ∈ ℝ, > 0 and ∼ N(0, 1).

ii) Briefly describe how your calculations and solution in (i) above would change if the

restriction > 0 is removed.

[8 marks]

(c) Briefly describe how independence of two jointly continuous random variables (, )

relates to properties of their conditional probability density functions.

[4 marks]

2. (a) Suppose that0 ∼ Poi() (where > 0 is fixed) and that for = 1, 2,… the distribution

of conditional on −1 = is Poisson with parameter .

Determine formulas for [] and Var() (for all = 0, 1, 2,… ). [10 marks]

(b) Suppose that and have joint probability density function

,(, ) = {

2e−−, > 0, 0 < < ,

0, otherwise

and define = and = . It follows (you do not have to prove this) that (, )

takes values in the set

= {(, ) ∈ ℝ2 ∶ > 0, ∈ (0, 2)}.

i) Find the joint probability density function of (, ).

ii) Determine the marginal probability density function of . Any integrals involved

should be evaluated as far as possiblewithout attempting to find any antiderivatives.

[10 marks]

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2 MATH2010-E1

3. (a) Suppose that the random variable has moment generating function

() =

e3

1 − 2

, ∈ (−1, 1).

i) Determine [].

ii) Suppose that ( = 1, 2, 3,… ) are independent and identially distributed copies

of and define

=

1

(1 + 2 +⋯+).

Find the moment generating function of and hence find the distribution of a

random variable that satisfies

→ .

[9 marks]

(b) Suppose that random variables 1 and 2 have joint moment generating function

1,2(1, 2) = 2(1 − 1 − 2)

− (2 − 1)

− 1 + 2 < 1, 1 < 2,

for some parameters > 0 and > 0.

Find the moment generating function of the random variable = 21+2, making

sure that the function is defined on a suitable domain. [4 marks]

(c) Suppose that = (1, 2, 3)

⊤ ∼ N3(, ), where

=

(

2

0

20)

and =

(

8 0 2

0 8

2 8)

.

For which values of the constants , ∈ ℝ is a valid covariance matrix; and when is

degenerate? [7 marks]

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SECTION B

4. Consider the discrete-time Markov chain { ∶ ≥ 0} on = {1, 2, 3, 4} with the transition

matrix

=

⎛

⎜

⎜

⎜

⎜

⎝

0 1 − 0

0 1 − 0

1++

0 1

1++

1++

0 0 1 −

⎞

⎟

⎟

⎟

⎟

⎠

where 0 ≤ ≤ 1 and 0 ≤ ≤ 1.

(a) In each of the following cases, classify each of the states as recurrent or transient.

i) = 1, = 1

2

;

ii) = 1

2

, = 0.

[4 marks]

Now assume that = 1

2

and = 1

3

.

(b) Find the equilibrium distribution for the chain. [4 marks]

(c) Determine the value of lim→∞ ( = 1 ∣ 0 = 1), stating clearly any results used.

[5 marks]

(d) Assume 0 = 1. By conditioning on the first step or otherwise, find the probability of

returning to state 1 before reaching state 4, stating all your reasoning clearly.

[7 marks]

MATH2010-E1

4 MATH2010-E1

5. (a) Consider the following Markov chain { ∶ ≥ 0} on the state space = {1, 2, 3}.

1 3

2

1

1

2

/

1

2

/

/

4

1

/

2

1

/

4

1

i) Find (6 = 1 ∣ 4 = 2, 0 = 1).

ii) Find ( = 2 for some ≥ 1 ∣ 0 = 2).

iii) Find the mean passage time to the set = {3}, starting from state 2.

[10 marks]

(b) Consider a simple symmetric random walk { ∶ ≥ 0} on ℤ.

i) Calculate the number of distinct paths for the random walk to evolve from 0 = −2

to 10 = 6.

ii) Given 0 = 0, find the probability that 10 = 2 and = 5 for some 0 < < 10.

iii) If we change the starting point0 to 10 in (ii), calculate the corresponding probability

in (ii) and comment on your result.

[10 marks]

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6. (a) i) Consider a branching process { ∶ ≥ 0} with 0 = 1 and offspring distribution

given by

=

⎧⎪

⎪

⎨

⎪

⎪⎩

0 with probability 1

3

1 with probability

2 with probability 2

3

− ,

where 0 ≤ < 2

3

. Find the extinction probability of the process, in terms of .

ii) Let represent the total progeny of the branching process in (i). Given that

[(

1

2)

]

=

2

9

,

find .

iii) Suppose now that the number 1 in the first generation has distribution

(1 = ) =

1

+ 1

, = 0, 1,… ,,

where is a fixed positive integer, and that the offspring distribution for the

remaining generations is as in (i). Find the extinction probability of the process.

[12 marks]

(b) Consider a renewal process, where the time between successive renewals has distribution

=

⎧⎪

⎨

⎪⎩

1 with probability 6

13

2 with probability 7

13

.

i) Show that

(a renewal takes place at time ) =

13

20

+

7

20 (

−

7

13)

.

ii) Imagine that the renewal process models the completion time of items on a production

line in a factory. Having started at time zero, a power cut occurs at time 6.25. Find

the probabilities that the product directly affected by the power cut has already

been under construction for 0.25 and 1.25 time units.

[8 marks]

MATH2010-E1 END