ASM Practice Exam 1 - problem 10

[edm17]

The memoryless property of the exponential distribution continues to elude me.

I get the parameter value of 707.39. I am thinking this value is the expected total loss for a claim since I solved for the ground up loss.

I am thinking the question asks for the expected loss per payment which equals

E(x) / (1 - F(500) )

However the solution just adds 500 to the 707 (which I thought is already the loss with the deductible included). It also doesn't divide by s(500).

Could someone please try and enlighten me on my logic errors and this concept. Thanks.

[Abraham Weishaus]

Look at it as a survivorship problem.

If there is a lack of memory, and average lifetime is 70 years, and you've lived 30 years already, your expected future lifetime is 70 years, not 40 years. If it were 40 years, then there would be a memory of having lived 30 years.

Here too, the fact that you're given that the claim is over 500 does not influence the average size of a claim. The average size is still 707, which means that if you count the amount you're not paying, the total size is 1207.

Memorylessness means that the conditional distribution is the same as the unconditional distribution. The distribution of the claim size minus 500, given that the claim is over 500, is the same as the distribution of the total claim size with no condition.

[edm17]

Thanks, that helps.

[eagle_halo]

But why the formula for mean excess loss doesnt work?

e(x)=(E(x)-E(X^x))/S(x)= (707 - 707(1-S(x)))/S(x) = 707

[Abraham Weishaus]

So it works (just add the 500 to the "excess"), but this is not the easiest way to do the problem.