# Information Matrix

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### Don't fear the integral!

Actuaries denote with $${}_tp_x$$ the probability that a life alive aged exactly $$x$$ years will survive a further $$t$$ years or more.  The most basic result in survival analysis is the following relationship with the instantaneous mortality hazard, $$\mu_x$$:

${}_tp_x = e^{-H_x(t)}\qquad(1)$

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

$H_x(t) = \int_0^t\mu_{x+s}ds\qquad(2).$

Written by: Stephen Richards

### Some points for integration

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.

Written by: Stephen Richards