Mgf Of Aggregate

[gcpgcp]

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 ?

 

[Busy_Bee4422]

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.

 

[Busy_Bee4422]

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.

 

[gcpgcp]

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.