SOA #93

[jrmkr]

In the solution, how X + Y = 8 became?

There was a graph so I unable to post it here.

The question is:

A family buys two policies from the same insurance company. Losses under the two policies are independent and have continuous uniform distributions on the interval from 0 to 10. One policy has a deductible of 1 and the other has a deductible of 2. The family experiences exactly one loss under each policy.

Calculate the probability that the total benefit paid to the family does not exceed 5.

[Gandalf]

The deductibles. If, for example, X=1 and Y=7, the payment is (X-1)+(Y-2)=5.

Note that if X<1 or Y<2, X+Y=8 is not the boundary.

[jrmkr]

(X-1)+(Y-2)=5.

Nice trick. Thanks.

If X<1 or Y<2, X+Y=8 is not the boundary.

It is true. But when integrating it is(the boundary) the limit as same as when X=1 and Y=2. Please could you explain it?

[Gandalf]

It is OK when X>=1 and Y>=2. It is only a problem if X<1 or Y<2, because if X<1 the benefit for X is 0, not X-1; similarly for Y.