Can someone advise why are we taking in account the conditional probabilities of winning 50 and 100 in the premium bond example towards the end of section 3.2 in Chapter 11.
The Xi's represent the amount of the ith win. So the amount you win given that you have won something. Hence you need to use the conditional probabilities of winning £50 or £100 given that you have won.
Hi Darren, If I was approaching this question without looking at the solution, my intuition wouldn't have been to have this conditional probability. I would have probably just let Xi's represent the individual winnings amount (i.e. it can be 0 as well then). But obviously, after looking at the solution, it makes it a lot easier to have that conditional probability. But I still can't understand how do we know to approach this question this way? Anything about any wording in this question that makes it obvious or am I just a bad student ? Thanks
Ah I see where you're coming from Qayanaat. Well let's try to construct a solution your way and see where it takes us ... Whenever you have any aggregate claims distribution, you should start by defining X, and N, and state those distributions. It really helps you keep on track. Here goes: Let Xi = the amount won on the i-th bond. (It's not necessarily a 'win', as you say.) The distribution of Xi is: P(Xi=0)=1-16/312,000 P(Xi=50)=15/312,000 P(Xi=100)=1/312,000. So far so easy. But what about N? Isn't this just the number of bonds? But that's a constant, 1,000. Not a random variable at all. So assumption 3.1 doesn't hold, and we can't use Panjer's formula. As a tip for the future, in general, try not think of the Xi's being zero. After all, the Collective Risk Model is all about adding up claim amounts, not zero amounts.
