I've been reviewing flashcards from the coachingactuary site. (Love the material and format, just ran out of time). There is a few flashcard that cover coinsurance, deductible, policy limit, etc. There are solution that a bit confusing to me in there difference.
4.1.13
For discrete loss amount X on a policy with deductible d, what is the formula for the expected uncovered loss, Z?
E[Z]={[Sum(z<=d)]x*Pr(X=x)+d*Pr(X>d)
4.2.8
For discrete loss amount X on a policy with deductible d and a policy limit u, what is the formula for the expected insurance payment, Y?
E[Y] = {[Sum(d<x<d+u)](x-d)*Pr(X=x)}+u*(Sx{d+u})
So the question for me lies with the last term in both solutions. Why is one denoted as d*Pr(X>d) and the other u*(Sx{d+u})? Specifically, why does one use a survival equation to determine the likely hood of being over the policy limit, and the other denote as simply the probability for being over the deductible? To my mind the the last top term could also be written as d*Sx(d) or the bottom equivalently u*Pr(x>d+u). Am I missing something?
4.1.13
For discrete loss amount X on a policy with deductible d, what is the formula for the expected uncovered loss, Z?
E[Z]={[Sum(z<=d)]x*Pr(X=x)+d*Pr(X>d)
4.2.8
For discrete loss amount X on a policy with deductible d and a policy limit u, what is the formula for the expected insurance payment, Y?
E[Y] = {[Sum(d<x<d+u)](x-d)*Pr(X=x)}+u*(Sx{d+u})
So the question for me lies with the last term in both solutions. Why is one denoted as d*Pr(X>d) and the other u*(Sx{d+u})? Specifically, why does one use a survival equation to determine the likely hood of being over the policy limit, and the other denote as simply the probability for being over the deductible? To my mind the the last top term could also be written as d*Sx(d) or the bottom equivalently u*Pr(x>d+u). Am I missing something?
Survival function vs Pr(x>u)