First we fix the first claim at time t. So, we say that it is definitely happening at time t.
The trick is that the only way we can be ruined is if the first claim is bigger than the amount we have in the bank at time t. We start with 2 and we receive premiums at rate 1.3. So by time t we will have 2+1.3t in the bank. We calculate the probability that a claim is bigger than this.
So, we know the answer if the first claim is definitely happening at time t. It is exp(-2+1.3Lt)
But when is the first claim? It could be at any time t, right?
I mean, if it's at time 4, the answer is exp(-2+1.3L*4)
I mean, if it's at time 7, the answer is exp(-2+1.3L*7)
I mean, if it's at time 27, the answer is exp(-2+1.3L*27)
OMG why can't someone tell me when the first claim is going to be and then I have an answer. Well, we don't know but we can tell you it is an exp(L) distribution, for which we know the PDF. So, we just condition on this.
We are effectively summing (integrating) across all possible values of t....
"P[first claim is at time t]" * exp(-2+1.3Lt)
Good luck!
John
