Actuarial Outpost Survival Function, not integrating from 0
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#1
01-11-2019, 04:24 PM
 p_tybg SOA Join Date: Nov 2018 Posts: 1
Survival Function, not integrating from 0

Question from ASM 22.25 chapter on Exponential Distribution.

An auto insurance policy has a deductible of 1 and a maximum claim payment of 5. Auto loss amounts follow an exponential distribution with mean 2. Calculate the expected claim payment made for an auto loss.

The solution says to integrate the survival function from 1 to 6, but the chapter on Mean says that the survival function must always be integrated from zero, even if the support does not include zero. Am I missing something?

Thanks

edit: From the ASM manual: "If a policy has a benefit limit, you may want to calculate the expected payment per loss, which is the expected value of the minimum of the loss random variable and the limit. To do that you integrate the survival function up to the limit."

Perhaps the reason we integrate from 1 to 6 is that we are interested in not the loss, but the payment— the payment is zero when the loss is 1 and 5 when the loss is 6. Thus we integrate the survival function from where loss in which the payment is zero to the loss in which the payment is five?

Last edited by p_tybg; 01-11-2019 at 04:33 PM.. Reason: Additional info
#2
01-11-2019, 04:43 PM
 Michael Mastroianni Member SOA Join Date: Jan 2018 Posts: 38

It’s because you aren’t calculating the mean of $X$ with the survival function method, but instead computing the mean of another random variable using the survival function of $X$.

So yes your edit is correct.
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Michael Mastroianni, ASA
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