# Information Matrix

## Filter 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

**Tags:**Filter information matrix by tag: survival curve, Filter information matrix by tag: integrated hazard function, Filter information matrix by tag: numerical integration

### 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

**Tags:**Filter information matrix by tag: life expectancy, Filter information matrix by tag: survival curve, Filter information matrix by tag: numerical integration, Filter information matrix by tag: adaptive quadrature, Filter information matrix by tag: Trapezoidal Rule, Filter information matrix by tag: Simpson's Rule