# 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

### Simulation and survival

In an earlier post we discussed how a survival model was directly equivalent to assuming future lifetime was a random variable. One consequence of this is that survival models make it quick and simple to simulate a policyholder's future lifetime for the purposes of ICAs and Solvency II.

**Written by:**Stephen Richards

**Tags:**Filter information matrix by tag: survival curve, Filter information matrix by tag: ICA, Filter information matrix by tag: Solvency II, Filter information matrix by tag: integrated hazard function