Transcript MAS 2317 Question 11

MAS 2317 Question 11
ANDREW MACGILL
Question
The number of defects in a 1200 foot roll of magnetic recording tape has a Po(u) distribution.
The prior distribution for u is Ga(3,1). When 5 rolls of this tape are selected at random and
inspected, the number of defects found on the rolls are 2, 2, 6, 0 and 3. Determine the posterior
distribution of u.
Posterior ∝ Prior X Likelihood
It has been made (VERY) clear that the posterior distribution is proportional to the product of
the prior distribution and the likelihood from the data collected. That is:
𝜋 𝜃 𝒙 ∝ 𝜋 𝜃 x 𝑓(𝒙|𝜃)
Where the posterior distribution is 𝜋 𝜃 𝒙 , the prior distribution is 𝜋 𝜃 and the likelihood from
the data that we have collected is 𝑓(𝒙|𝜃).
Posterior ∝ Prior X Likelihood
Prior Distribution: Ga(3,1)
𝛽 𝛼 𝜃 𝛼−1 𝑒 −𝛽𝜃
P.d.f =
𝛤(𝛼)
=
13 𝜃2 𝑒 −𝜃
𝛤(3)
Posterior ∝ Prior X Likelihood
Likelihood: Po(u)
P.m.f = (uke-u)/k!
𝑓(𝒙|𝜃)
=
𝜃
𝑛
𝑖=1(
𝑘 −𝜃
𝑒
𝑘!
Inputting the data gives:
◦=
𝜃13 𝑒 −5𝜃
17280
)
𝜃 2 𝑒 −𝜃
2!
X
𝜃 2 𝑒 −𝜃
2!
X
𝜃 6 𝑒 −𝜃
6!
𝜃 0 𝑒 −𝜃
X
0!
X
𝜃 3 𝑒 −𝜃
3!
Posterior ∝ Prior X Likelihood
Combining the two gives:
𝜋 𝜃 𝒙 ∝ 𝜋 𝜃 x 𝑓(𝒙|𝜃)
𝜋 𝜃𝒙 ∝
13 𝜃 2 𝑒 −𝜃
𝛤(3)
x
𝜃 13 𝑒 −5𝜃
17280
= 𝑘𝜃13 𝜃 2 𝑒 −5𝜃 𝑒 −𝜃 ,
= 𝑘𝜃15 𝑒 −6𝜃
◦ ∴ 𝜋 𝜃 𝒙 ~𝐺𝑎(16,6)
where k is a constant.
Thanks for listening