Deductibles still make no sense

Problem 1:



For an insurance:



I. Loss amounts are uniformly distributed on the interval [0,12]

II. There is an ordinary deductible of 3 per loss.



Determine the 90th percentile of the insurance payment.



a) 7.80

b) 7.30

c) 0

d) 6.80

e) 8.00



I rationalized it a ton of different ways and always got the wrong answer, and finally got it right when it clicked that we can't just disregard a payment of 0 (ie, the lower 25%), because we in fact PAY NOTHING 25% OF THE TIME - key distinction



So all is well and dandy, then I come to this problem, problem 2:



Let X and Y be random losses with joint density function 2x for 0 < x < 1 and 0 < y < 1. An insurance policy is written to cover the loss X + Y and has a deductible of 1. Calculate the expected payment under the policy.



So I integrate over the correct bounds of the square region the density is defined, and get 1/4. Then I rationalize it as 'well, 50% of the time we pay this, as 50% of the time we're in this region. The other 50% of the time we aren't and haven't met the deductible. Therefore we're only paying this 1/4 50% of the time, and the answer is 1/8'



Nope, as usual it's wrong



So in the first problem the logic we use works, in the second problem the logic we use doesn't work, and I'm back to understanding nothing about these kinds of problems

Deductibles still make no sense