- 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 tofailure rate , also calledhazard function , inreliability 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 theconditional probability of a person, who attained age ("x"), dying from age ("x") to age ("x"+"Δx") as:$P(x+delta;xmid;x>x)=frac\{F\_X(x+Delta;x)-F\_X(x)\}\{Delta;x(1-F\_X(x))\}$

where F

_{X}(x) is thedistribution function of the continuous age-at-deathrandom variable , X. If we let "Δx" tend to zero, we get a function for**force of mortality**, denoted as "μ(x)":$mu,(x)=frac\{F\text{'}\_X(x)\}\{1-F\_X(x)\}$

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:$mu,(x)=frac\{f\_X(x)\}\{1-F\_X(x)\}=-frac\{s\text{'}(x)\}\{s(x)\}=-\{frac\{d\}\{dxln\; [s(x)]$

**See also***

Failure rate

*Hazard function

*Actuarial present value

*Actuarial science

*Reliability theory **External links***http://www.fenews.com/fen46/topics_act_analysis/topics-act-analysis.htm

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