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MATH6174 Likelihood and Bayesian Inference

创建日期：2024-04-24 16:30:37

所属分类：数学mathematics

标签： MATH6174

Likelihood and Bayesian Inference

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MATH6174 Likelihood and Bayesian Inference
Duration: 2.5 hours (2 hours to complete, 30 minutes to upload)
This paper contains four questions.
Answer ALL questions.
An outline marking scheme is shown in brackets for each question.
All students:
This final assessment is to be submitted via Blackboard. The availability window for this
assessment is 8 hours. The availability window starts at 9am and ends at 5pm. You need
to have submitted your work by 5pm at the latest. You will not receive marks for work
uploaded after 5pm.
The expected working time to complete this assessment is 2 hours.
Once you download the paper, you have 2 hours to complete your work. You will then
have 30 minutes to upload your paper. Please make sure you leave enough time for the
document to upload within the availability window and that you start your final upload at
least 10 minutes before the assessment closes.
Papers submitted after 2 hours 30 minutes will not recieve any marks.
It is your responsibility to upload the correct file(s) to the correct link. No marks will be
given for the assessment if the wrong file is uploaded
If you experience issues
... with your internet access
you could take screenshots of your computer screen or of your internet service provider’s
status. If possible, ensure that photographs include the time they were taken. These
should be submitted with a request for Special Consideration.

... during file upload within the availability window

If you do not upload your file within the availability window for the assessment, you should
send it as soon as possible after the assessment to [email protected],
[email protected] and [email protected] together with any sup-
porting information to explain why it is late and a request for Special Consideration. The
Special Considerations Board will decide whether or not to recommend to the Board of
Examiners that they accept and mark your late work.
Note: No academic enquiries will be answered by staff and no amendments to papers will
be issued during the examination. If you believe there is a misprint/typoe, note it in your
submission but answer the question as written.
This is an open book assessment. You may consult books, notes or internet sources. You
are permitted to use calculators or mathematical software, but to obtain full marks, you
must show and explain your working as well as the final answer.
The assessment must be carried out in accordance with the University Academic Integrity
regulations. It is not permitted to communicate with anyone else (be it private or online)
about the content of this exam during the whole time it is open.
• Start a new question on a fresh sheet of paper.
• Make sure your page is in portrait orientation.
• Write in Black or Blue pen.
• On each page, write your page number in the top left and your module and student ID
in the top right.
• Show all working
2 MATH6174W1
1. [25 marks] Assume that, given Λ = λ, the random variable X has probability
density function (p.d.f.)
fX(x|λ) = λ exp(−λx), x > 0, λ > 0,
i.e., X|Λ = λ ∼ exponential(λ). Further, assume that the random variable Λ has
p.d.f.
fΛ(λ) =
δα
Γ(α)
λα−1e−δλ, x > 0, α > 0, δ > 0,
i.e., Λ ∼ gamma(α, δ).
(a) [5 marks] Write down the joint p.d.f. of X and Λ, and show that the marginal
p.d.f. of X is
fX(x) =
αδα
(x+ δ)α+1
, x > 0, α > 0, δ > 0.
(b) [8 marks] Show that
E(X) =
δ
α− 1 and E(X
2) =
2δ2
(α− 1)(α− 2),
and clearly state for what values of α and δ these moments exist.
(c) [2 Marks] Find the variance of X .
(d) [6 marks] If X1, . . . , Xn are independent, identically distributed random
variables with p.d.f. fX(x) given in part (a), then find the method of moments
estimators of α and δ.
(e) [4 marks] Find the p.d.f. of Z = X2 and E(Z).
Copyright 2021 © University of Southampton Page 2 of 5
3 MATH6174W1
2. [25 marks] Assume that y1, . . . , yn are observations of Y1, . . . , Yn, where Yi,
i = 1, . . . , n, are independent, normally distributed random variables with mean
α + βxi and unknown variance σ2, and x1, . . . , xn are known constants such that∑n
i=1 xi = 0.
(a) [11 marks] Show that the maximum likelihood estimates of α, β and σ2 are
given by
αˆ = y¯,
βˆ =
∑n
i=1 xiyi∑n
i=1 x
2
i
,
σˆ2 =
1
n
n∑
i=1
(yi − αˆ− βˆxi)2,
where y¯ = 1n
∑n
i=1 yi.
(b) [4 marks] Show that the maximum likelihood estimators for α and β are
unbiased and find their variances.
(c) [3 marks] Show that the Fisher information matrix for θ = (α, β, σ2)T is
I(θ) =

n
σ2 0 0
0
∑n
i=1 x
2
i
σ2 0
0 0 n2(σ2)2
 .
(d) [2 marks] What are the Crame´r–Rao lower bounds for unbiased estimators of α
and β? Do the maximum likelihood estimators for α and β attain their bounds?
(e) [2 marks] What is the asymptotic distribution of σˆ2?
(f) [3 marks] Show that (
σˆ2
1 + 1.96
√
2/n
,
σˆ2
1− 1.96√2/n
)
is an approximate 95% confidence interval for σ2.

TURN OVER
Page 3 of 5
4 MATH6174W1
3. [25 marks] Suppose that Y1, . . . , Yn are independent random variables, each having
probability density function f(y|θ) = θyθ−1 with 0 < y < 1. Assume the gamma
prior distribution pi(θ) = β

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