# Force of mortality

Force of mortality

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 "qx". 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"+"&Delta;x") as:

where FX(x) is the distribution function of the continuous age-at-death random variable, X. If we let "&Delta;x" tend to zero, we get a function for force of mortality, denoted as "&mu;(x)":

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

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

$mu,\left(x\right)=frac\left\{f_X\left(x\right)\right\}\left\{1-F_X\left(x\right)\right\}=-frac\left\{s\text{'}\left(x\right)\right\}\left\{s\left(x\right)\right\}=-\left\{frac\left\{d\right\}\left\{dxln \left[s\left(x\right)\right]$

* Failure rate
* Hazard function
* Actuarial present value
* Actuarial science
* Reliability theory

*http://www.fenews.com/fen46/topics_act_analysis/topics-act-analysis.htm

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