I got this problem and I've seen the solution but I'm thinking there's a better way to do it though I can't quite get it.
A company insures 2 machines for maintenance. In a given year, each machine will require maintenance with probability 1/3, and if maintenance is required, the cost will be a uniform random variable between 0 and 4000. If the insurance policy has an annual payment limit of 6000 for both machines combined, what is the expected annual payment made by the insurer? (Answer: 1325)
The solution did it using the survival function with some pretty messy results. Is there a way to break this up into cases and just do something like
E(payment) = E(payment | case 1) P(case 1) + ...+ E(payment |case n)P(case n)?
I tried doing something like that but it didn't work. I tried breaking it up in the following way:
0 if x,y=0
X if x>0, y=0
Y if x=0, y>0
X+Y if X>0, Y>0, X+Y<=6000
6000 if X>0, Y>0, X+Y>6000
I tried it like this and did E(payment|x,y=0)*P(x,y=0) + E(payment|x>0, y=0)+...and so on.
When you do E(payment | x>0, y=0)*P(x>0, y=0), can you do:
E(x)*(1/3)(2/3)? Or is that wrong?
Is it possible to do the problem the way I outlined? How else can you do it without the survival function?