Actuarial Outpost SOA #87
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#1
12-21-2018, 01:10 PM
 joelcheung SOA Join Date: Mar 2018 College: Singapore Management University, sophomore Posts: 5
SOA #87

Looking at the 6th line in the solution, i.e. E(X-20)+ = E(X) - ....

I understand that this line is using the formula E[(X-20)+] = E(X) - E[(X^20)],
but how does that second part of the 6th line equate to E[(X^20)]??

And for this particular question, can I use the survival function (derived by finding the distribution function F(x) from the f(x) in the solution and then S(x)= 1 - F(x)), and then integrating the survival function from 20 to 80 for the flat f(x) portion + integrating the survival function from 82 to 120 for the declining f(x) portion?

Thank you!
#2
12-21-2018, 01:15 PM
 daaaave David Revelle Join Date: Feb 2006 Posts: 3,037

Whenever X is a continuous loss variable,
$E[X \wedge 20] = \int_0^{20} x \cdot f(x) \, dx + 20 \Pr[X>20]$
In their expression, they are finding Pr[X>20] by doing 1 - Pr[X<=20] as the integral for P[X<=20] is easier than the integral for Pr[X>20].

Yes, you could use the survival function. I think it will take longer.
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