# Information Matrix

## Filter Information matrix

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### Introducing the Product Integral

Of all the actuary's standard formulae derived from the life table, none is more important in survival modelling than:

\[{}_tp_x = \exp\left(-\int_0^t\mu_{s+s}ds\right).\qquad(1)\]

**Written by:**Angus Macdonald

**Tags:**Filter information matrix by tag: survival models, Filter information matrix by tag: survival probability, Filter information matrix by tag: force of mortality, Filter information matrix by tag: product integral

### From small steps to big results

In survival-model work there is a fundamental relationship between the \(t\)-year survival probability from age \(x\), \({}_tp_x\), and the force of mortality, \(\mu_x\):

\[{}_tp_x = \exp\left(-\int_0^t\mu_{x+s}ds\right).\qquad(1)\]

**Written by:**Stephen Richards

**Tags:**Filter information matrix by tag: survival probability, Filter information matrix by tag: force of mortality, Filter information matrix by tag: differential equation