### Some points for integration

#### (Mar 2, 2016)

The survivor function from age $$x$$ to age $$x+t$$, denoted $${}_tp_x$$ by actuaries, is a useful tool in mortality work.  As mentioned in one of our earliest blogs, a basic feature is that the expected time lived is the area under the survival curve, i.e. the integral of $${}_tp_x$$.  This is easy to express in visual terms, but it often requires numerical integration if there is no closed-form expression for the integral of the survival curve.  In this article we look at some of the options available to actuaries who need to integrate numerically.

A general result is that the survivor function has the following form:

${}_tp_x = e^{-H_x(t)}$

where $$H_x(t)$$ is the integrated hazard function:

\[H_x(t) = \int_0^t…