April 02 Qu 7) i)

Yes - use the ASET or revision notes from ActEd with clearer solutions

This question is slightly different to the standard compound distribution questions because we actually have a mixture distribution.

For the number of claims we have:

N | lambda = Poi(lambda)

where

lambda ~ gamma (2, 10)

So we need to use the formula at the top of page 16 of the Tables to get the mean and variance of N so we can then use that to get the mean and variance of S using the compound distribution formulae also on page 16 of the Tables.

So here we go:

E(N) = E[E(N|lambda)] = E[lambda] = 2/10 = 0.2

var(N) = E[var(N|lambda)] + var[E(N|lambda)]
= E[lambda] + var[lambda]
= 2/10 + 2/10² = 0.22

So now we can use:

E(S) = E(N)E(X) and var(S) = E(N)var(X) + E²(X)var(N)

 

Hmmm, good point. One of my students contacted admin and asked if it was possible to purchase the older CT6 ASET and she got it - so it must be possible. Do give them a ring on 01235 550005 or email acted@bpp.com if you want to get an older version.

 

At the very end we do add up 100 of these policies - but be careful with your variance:

var(nX) = n²var(X)

but

var(X1 + ...+ Xn) = nvar(X)

so it will actually be:

var(S) = 100 × ....