From small steps to big results

(Feb 1, 2018)

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, $$\mu_x$$:

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

Where does this relationship come from?  We start by extending the survival time by an amount, $$h$$, and look at the $$(t+h)$$-year survival probability:

${}_{t+h}p_x = {}_tp_x.{}_hp_{x+t}\qquad(2)$

which is simply to say that in order to survive $$(t+h)$$ years, you first need to survive $$t$$ years and then you need to survive a further $$h$$ years.  Of course, surviving $$h$$ years is the same as not dying in $$h$$ years, so equation (2) can be written thus:

\[{}_{t+h}p_x…