We all know the integration by parts formula: though, of course, there is some choice in what is; any anti-derivative will do. Well, sort of.

I thought about this as I’ve been roped into teaching an actuarial mathematics class (and no, I have zero training in this area…grrr…)

So here is the set up: let where is the random variable that denotes the number of years longer a person aged will live. Of course, is a probability distribution function with density function and if we assume that is smooth and has a finite expected value we can do the following: and, in principle this integral can be done by parts….but…if we use we have:

\

which is a big problem on many levels. For one, and so the new integral does not converge..and the first term doesn’t either.

But if, for we note that is the survival function whose limit does go to zero, and there is usually the assumption that as

So we now have: which is one of the more important formulas.