Force of mortality

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In actuarial science, force of mortality represents the instantaneous rate of mortality at a certain age measured on an annualized basis. It is identical in concept to failure rate, also called hazard function, in reliability theory.

In a life table, we consider the probability of a person dying from age (x) to (x+1), called q_x. In the continuous case, we could also consider the conditional probability of a person, who attained age (x), dying from age (x) to age (x+Δx) as:

$P(x<X<x+\Delta\;x\mid\;X>x)=\frac{F_X(x+\Delta\;x)-F_X(x)}{1-F_X(x)}$

where F_X(x) is the distribution function of the continuous age-at-death random variable, X. If we let Δx tend to zero, we get a function for force of mortality, denoted as μ(x):

$\mu\,(x)=\frac{F'_X(x)}{1-F_X(x)}$

Since f_X(x)=F'_X(x) is the probability density function of X, and s(x)=1-F_X(x) is the survival function, force of mortality can also be expressed variously as:

$\mu\,(x)=\frac{f_X(x)}{1-F_X(x)}=-\frac{s'(x)}{s(x)}=-{\frac{d}{dx}}ln[s(x)]$