• We are pleased to announce that the winner of our Feedback Prize Draw for the Winter 2024-25 session and winning £150 of gift vouchers is Zhao Liang Tay. Congratulations to Zhao Liang. If you fancy winning £150 worth of gift vouchers (from a major UK store) for the Summer 2025 exam sitting for just a few minutes of your time throughout the session, please see our website at https://www.acted.co.uk/further-info.html?pat=feedback#feedback-prize for more information on how you can make sure your name is included in the draw at the end of the session.
  • Please be advised that the SP1, SP5 and SP7 X1 deadline is the 14th July and not the 17th June as first stated. Please accept out apologies for any confusion caused.

Mgf Of Aggregate

G

gcpgcp

Member
Q: In a group of policies, the monthly number of claims for a single policy has a Poisson distribution with parameter λ, where λ is a random variable with the following density: f(λ) = 2 exp { -2 λ } ; λ > 0
Find the moment generating function for the aggregate claims distribution if the claims have a gamma distribution with mean 2 and variance 2.

The solution uses a method for calculation which is greek to me (see attachment).

a) Can I use Ms(t) = Mn (log Mx(t)) ? Please explain steps to solve if YES.

b) If NO , Why ?

c) Where to find the method used for solving (as in attachment) in the core reading material ?
 

Attachments

  • 1.pdf
    122.7 KB · Views: 201
Q: In a group of policies, the monthly number of claims for a single policy has a Poisson distribution with parameter λ, where λ is a random variable with the following density: f(λ) = 2 exp { -2 λ } ; λ > 0
Find the moment generating function for the aggregate claims distribution if the claims have a gamma distribution with mean 2 and variance 2.

The solution uses a method for calculation which is greek to me (see attachment).

a) Can I use Ms(t) = Mn (log Mx(t)) ? Please explain steps to solve if YES.
You could use the formula. The challenge here is what is the MGF of the number of claims given that f(n) = 2/[3^(n+1)] for n = 0, 1, 2, .... That is what the answer is trying to avoid calculating.

b) If NO , Why ?

c) Where to find the method used for solving (as in attachment) in the core reading material ?

This is not dealt with directly in the notes given that this is simply about mathematical manipulation of MGFs and PGFs as covered in CT3.
 
Explanation of Solution

Ms(t) = E[e^(St)] = E{E[e^(St) |N]}

= E[{E[e^(Xt)]}^N]

This is the first steps in the acted notes on the MGF of an aggregate model.

= Gn[Mx(t)]

This follows because from the PGF formula

Since {E[e^(Xt)]}^N = {Mx(t)}^N

If we substitute Mx(t) by z

E[{E[e^(Xt)]}^N] = E(z^N) resembling the formula for a PGF

Hence

E[{E[e^(Xt)]}^N] = Gn[Mx(t)]

The PGF route was chosen because it is easier to calculate than the MGF

The rest of it is just mathematical manipulation to get the PGF for N and then eventually the MGF of the aggregate model.
 
The first part is easy.

The second part stumped me.

They have used: Gx(t) = E[t^X] = Summation { t^X . P[N=n]}
instead of Ms(t) = Mn(t) ......

Thanks for your reply.
 
Back
Top