UNIVERSITY OF SOUTHAMPTON MATH6174W1
SEMESTER 1 EXAMINATION 2020/21
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.
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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).
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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
α = yˉ,
β =
∑n
i=1 xiyi∑n
i=1 x
2
i
,
σ 2 =
1
n
n∑
i=1
(yi α β xi)2,
where yˉ = 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) =
