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Examination Paper for STAT0005 Page 1

STAT0005: Probability and Inference

2019/20, Level 5

Answer ALL questions.

You may submit only one answer to each question.

The relative weights attached to each section are Section A (39 marks),

Section B (59 marks), Section C (38 marks).

The numbers in square brackets indicate the relative weights attached

to each part question.

Marks are awarded not only for the final result but also for the clarity

of your answer.

Asking questions during the coursework period

You may not email the course organizer directly during the period set

for this coursework.

If you need to clarify any part of the coursework, you may post to

the course Moodle forum within the first two working days of the

coursework’s release. No clarification questions will be answered after

this.

You may not ask questions other than clarifications at any point

during the period set for the coursework.

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Examination Paper for STAT0005 Page 2

Formatting your solutions for submission

Some part-questions require you to type your answers instead of handwriting

them. These questions state [Type] at the start of the partquestion.

You must follow this instruction. Failure to do so may

result in marks being deducted. For questions without the [Type]

instruction, you may choose to type or hand-write your answer.

Some part-questions ask you for an explanation using only words and

no formulae. If you use formulae anyway, these may be entirely ignored

in the marking process.

Where a word limit has been set for a part question, this has been

chosen to be at least three times the length of the expected answer.

Hence you should view the word limit as a strict upper limit rather

than as the number of words to achieve.

You should submit ONE document that contains your solutions for all

questions/ part-questions. Please follow UCL’s guidance on combining

text and photographed/ scanned work.

Make sure that your handwritten solutions are clear and are readable

in the document you submit. You are encouraged to write out solutions

neatly once you are happy with them.

Plagiarism and collusion

You must work alone. In particular, any discussion of the coursework

with anyone else is not acceptable. You are encouraged to read the

Department of Statistical Science’s advice on collusion and plagiarism,

which you can find here.

Parts of your submission will be screened via Turnitin to check for

plagiarism and collusion.

If there is any doubt as to whether the solutions you submit are entirely

your own work you may be required to participate in an investigatory

viva to establish authorship.

Continued

Examination Paper for STAT0005 Page 3

Section A

A1 Let X ∼ U(−1, 1) and let Y = X4

.

(a) Compute E[Y ]. [3]

(b) Compute Var(Y ). [3]

(c) Compute the pdf fY of Y . [5]

A2 Let the joint distribution of X and Y be given by the following two-way

table:

X

Y -1 0 1

-1 b 0 a

0 0 1-2a-2b 0

1 a 0 b

Here, a, b are unknown constants such that the above table is a valid

two-way table. For parts (a)-(d) you may leave your results in terms

of a and b where necessary.

(a) Compute the marginal pmfs pX and pY . [3]

(b) Compute E[X] and E[Y ]. [3]

(c) Compute Var(X) and Var(Y ). [3]

(d) Compute Corr(X, Y ) in the case (a, b) 6= (0, 0). [5]

(e) What constraints must the pair (a, b) satisfy to ensure that the

table above is valid? [2]

(f) What is the smallest value Corr(X, Y ) can take in this case? Give

a value of (a, b) for which the smallest possible value of Corr(X, Y )

is attained. [2]

A3 Let the joint cdf of X and Y be given by

FX,Y (x, y) =0 if x < 0 or y < 0

min{x, y} if x, y ≥ 0 and (x ≤ 1 or y ≤ 1)

1 if x, y ≥ 1.

(a) Compute P(0 < X ≤ 1, 0 < Y ≤ 1). [2]

(b) Compute the marginal cdf FY of Y . [3]

(c) Compute P(X < 1/2 | Y < 1/2).

[Type] Using only words and no formulae, decide whether X and

Y are independent and justify your decision. Maximum length:

150 words. [5]

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Examination Paper for STAT0005 Page 4

Section B

B1 For i ∈ {1, . . . , n} with n ∈ N and n ≥ 2, let Xi

i.i.d. ∼ N(µ, σ2) where µ

and σ

2 > 0 are both unknown. To estimate the variance σ2, consider

the estimator. In this expression, the number α ∈ (−∞, 2) is

used to obtain different estimators.

(a) Name the estimator in the case α = 1. What is the expected

value of T1? (You do not need to compute the expected value if

you know it.) [2]

(b) Compute the bias of Tα. [2]

(c) Compute the sampling variance of Tα. Hint: Start from the

variance of T1. [5]

(d) Show that the mse for Tα is given by

mse(Tα; σ2) = σ4(n − α)2α2 − 2α − 1 + 2n.

[4]

(e) Given the sample size n, which value α∗

(n) of α results in the

smallest mean square error of Tα? You need to provide a derivation

and justification of your result and while you may omit checking

the second order condition you should check the boundaries. [8]

(f) Hence provide a formula for an estimator of the form Tα with

smallest mse.

[Type] Using only words and no formulae, give one reason why T1

is often used in practice in spite of your result. Maximum length:

300 words. [5]

Continued

Examination Paper for STAT0005 Page 5

B2 For n ∈ N, consider the regression model of the form

Y = αx + , ∼ N(0, Σ),

where the covariate x ∈ R

n with x 6= 0 and the positive definite symmetric

matrix Σ ∈ R

n×n are fixed and known whereas the parameter

α ∈ R is unknown. The sample consists of a single observation y from

this model.

(a) [Type] Using only words and no formulae, explain how a maximum

likelihood estimator is obtained in general. Maximum length:

150 words. [4]

(b) In the setting described above, obtain the log-likelihood for the

parameter α given the one observation y ∈ R

n. [4]

(c) Show that the MLE of α is given by

αbMLE =xTΣ−1yxTΣ−1x.

You need to check all applicable conditions for a maximum.

Hint: Note that α is a number (not a matrix or a vector) and

note the dimension of x

TΣ−1

y as well as that of xTΣ−1x. [8]

(d) Compute E[αbMLE]. [5]

(e) Compute the Cramer-Rao lower bound for unbiased estimators of

α. What is the interpretation of this bound? [4]

(f) Compute the sampling variance of αbMLE. Hence decide whether

or not αbMLE achieves the Cramer-Rao lower bound.

Hint: First show that αbMLE is of the form αbMLE = c + a

T

for

some constant c ∈ R and some constant vector a ∈ R

n which you

should specify.

[8]

Turn Over

Examination Paper for STAT0005 Page 6

Section C

Let X ∼ N(µ, Σ) where µ ∈ R

n and Σ ∈ R

n×n

for dimension n ≥ 2.

Also assume that Σ is positive definite symmetric. Let A ∈ R

k×n with

k ∈ {1, . . . , n}. You may use the facts that, firstly, AΣAT

is positive definite

symmetric if A has full rank and that, secondly, A has full rank if and only

if its row vectors are linearly independent.

(a) Compute the mgf of Y = AX and thus show that Y follows a normal

distribution and specify its mean vector and covariance matrix. [4]

(b) Under the condition that A has full rank, write down the pdf of Y .

Explain why it is impossible to write down the pdf of Y if A does not

have full rank. [4]

(c) Using the mgf or otherwise, prove that Cov(Xi

, Xj ) = 0 if and only if

Xi and Xj are independent. Note that during the course we have only

established this in the case of bivariate normal distributions. [6]

(d) Suppose that Cov(X1, Xj ) = 0 holds for all j ∈ {2, . . . , d}. Show that

X1 is independent of X2 + X3 + . . . + Xd. [5]

(e) Consider three discrete random variables U, V, W each taking values in

{−1, 1} and such that U is independent of V and U is independent of

W. For each of the statements (i) and (ii) below, decide whether it

is true in general or whether it may be false. If the statement is true

in general, provide a proof. Otherwise, find a joint pmf for U, V and

W that provides a counterexample (i.e. a joint distribution for which

the statement does not hold), explaining your reasoning and presenting

your joint pmf in a table as follows:

u v w pUV W (u, v, w)

(i) U and V + W are uncorrelated. [4]

(ii) U and V + W are independent. [5]

(f) [Type] Using only words and no formulae, write a short essay on three

ways in which the Gaussian distribution is special among probability

distributions. You need to make clear how the ways are directly related

to this Section C and/or to results in the lecture notes. Maximum

length: 300 words. [10]

End of Paper

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