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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.
Turn Over
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]
Turn Over
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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