sol4.pdf - MTH5131 Actuarial Statistics Coursework 4 —...

MTH5131 Actuarial Statistics
Coursework 4 — Solutions
Exercise 1. 1. Let X be the number of claims received in a week. To determine the posterior
distribution of μ , we must calculate the conditional probabilities P ( μ = 8 | X = 7) , P ( μ =
10 | X = 7) , and P ( μ = 12 | X = 7) ,. The first of these is
P ( μ = 8 | X = 7) =
P ( μ = 8 , X = 7)
P ( X = 7)
=
P ( X = 7 | μ = 8) P ( μ = 8)
P ( X = 7)
Since X ∼ Poisson( μ ) ,
P ( X = 7 | μ = 8) =
e - 8 8 7
7!
and since the prior distribution is uniform on the integers 8, 10 and 12:
P ( μ = 8) =
1
3
.
So
P ( μ = 8 | X = 7) =
e
- 8
8
7
7!
× 1
3
P ( X = 7)
=
0 . 04653
P ( X = 7)
Similarly,
P ( μ = 10 | X = 7) =
P ( X = 7 | μ = 10) P ( μ = 10)
P ( X = 7)
=
e
- 10
10
7
7!
×
1
3
P ( X = 7)
=
0 . 03003
P ( X = 7)
and
P ( μ = 12 | X = 7) =
P ( X = 7 | μ = 12) P ( μ = 12)
P ( X = 7)
=
e - 12 12 7
7!
×
1
3
P ( X = 7)
=
0 . 01456
P ( X = 7)
Since these conditional probabilities must sum to 1, the denominator must be the sum of the
numerators, so
P ( X = 7) = 0 . 04653 + 0 . 03003 + 0 . 01456 = 0 . 09112
The posterior probabilities are:
P ( μ = 8 | X = 7) =
0 . 04653
0 . 09112
= 0 . 51066
P ( μ = 10 | X = 7) =
0 . 03003
0 . 09112
= 0 . 32954
P ( μ = 12 | X = 7) =
0 . 01456
0 . 09112
= 0 . 15980
2. The Bayesian estimate under squared error loss is the mean of the posterior distribution:
8 × 0 . 51066 + 10 × 0 . 32954 + 12 × 0 . 15980 = 9 . 29830
1

Exercise 2. 1. Since the prior distribution of p is Beta(4,4):
f ( p ) ∝ p
3
(1 - p )
3
Now let X denote the number of successes from a sample of size n . Then X ∼ Binomial( n, p ) .
Since x successes have been observed, the likelihood function is:
L ( p ) = P ( X = x ) =
n
x
p
x
(1 - p )
n - x
∝ p
x
(1 - p )
n - x
Combining the prior PDF with the likelihood function gives:
f ( p | x ) ∝ p
3
(1 - p )
3
× p
x
(1 - p )
n - x
= p
x +3
(1 - p )
n - x +3
Comparing this with the PDF of the Beta distribution, we see that the posterior distribution
of p is Beta ( x + 4 , n - x + 4) .
The Bayesian estimate under all-or-nothing loss is the mode of the posterior distribution, ie the
value of p that maximises the posterior PDF. To find the mode, we need to differentiate the PDF
(or equivalently differentiate the log of the PDF) and equate it to zero.
Given that x = 10 and n = 25 , the posterior of p is Beta (15,18) and:
f ( p | x ) = Cp
14
(1 - p )
17
.