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| In [[actuarial science]], '''force of mortality''' represents the instantaneous [[Mortality rate|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]].
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| ==Motivation and definition==
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| In a [[life table]], we consider the probability of a person dying from age ''x'' to ''x'' + 1, called ''q''<sub>''x''</sub>. In the continuous case, we could also consider the [[conditional probability]] of a person who has attained age (''x'') dying between ages ''x'' and ''x'' + ''Δx'', which is
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| :<math>P_{x}(\Delta x)=P(x<X<x+\Delta\;x\mid\;X>x)=\frac{F_X(x+\Delta\;x)-F_X(x)}{(1-F_X(x))}</math>
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| where F<sub>X</sub>(x) is the [[distribution function]] of the continuous age-at-death [[random variable]], X. As ''Δx'' tends to zero, so does this probability in the continuous case. The approximate force of mortality is this probability divided by ''Δx''. If we let ''Δx'' tend to zero, we get the function for '''force of mortality''', denoted by <math>\mu(x)</math>: | |
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| :<math>\mu\,(x)=\frac{F'_X(x)}{1-F_X(x)}</math>
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| Since ''f''<sub>''X''</sub>(''x'')=''F'' '<sub>''X''</sub>(''x'') is the probability density function of ''X'', and ''S''(''x'') = 1 - ''F''<sub>''X''</sub>(''x'') is the [[survival function]], the force of mortality can also be expressed variously as:
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| :<math>\mu\,(x)=\frac{f_X(x)}{1-F_X(x)}=-\frac{S'(x)}{S(x)}=-{\frac{d}{dx}}ln[S(x)].</math>
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| To understand conceptually how the force of mortality operates within a population, consider that the ages, ''x'', where the probability density function ''f''<sub>''X''</sub>(''x'') is zero, there is no chance of dying. Thus the force of mortality at these ages is zero.
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| The force of mortality <math>\mu(x)</math> can be interpreted as the ''conditional'' density of failure at age ''x'', while ''f''(''x'') is the ''unconditional'' density of failure at age ''x''.<ref name=MQR>R. Cunningham, T. Herzog, R. London (2008). ''Models for Quantifying Risk, 3rd Edition'', Actex.</ref> The unconditional density of failure at age ''x'' is the product of the probability of survival to age ''x'', and the conditional density of failure at age ''x'', given survival to age ''x''.
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| This is expressed in symbols as
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| :<math>\,\mu(x)S(x) = f_X(x)</math>
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| or equivalently
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| :<math>\mu(x) = \frac{f_X(x)}{S(x)}.</math> | |
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| In many instances, it is also desirable to determine the survival probability function when the force of mortality is known. To do this, integrate the force of mortality over the interval ''x'' to ''x + t''
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| :<math> \int_{x}^{x+t} -\frac{d}{dy} ln[S(y)]\, dy </math>. | |
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| By the fundamental theorem of calculus, this is simply
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| :<math> ln[S(x + t)] - ln[S(x)], </math> | |
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| and taking the exponent to the base ''e'' results in
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| :<math> \frac{S(x+t)}{S(x)} = S(t). </math>
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| Therefore, the survival probability of an individual of age ''x'' is written in terms of the force of mortality as
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| :<math> S_x(t) = e^{-\int_x^{x+t}\mu(y)\, dy\,}. </math>
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| ==Examples==
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| This example is taken from.<ref>Dickson, David C.M., Cambridge (2009). ''Actuarial Mathematics for Life Contingent Risks, First Edition'', Cambridge University Press.</ref> A survival model follows Makeham's law if the force of mortality is
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| :<math>\mu(y) = A + Bc^{y} \quad\text{for } y \geqslant 0.</math>
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| Using the last formula, we have
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| :<math>\int_x^{x+t} A + Bc^{y} dy = At + B(c^{x+t}-c^{x})/ln[c].</math>
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| Then
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| :<math>S_x(t) = e^{-(At + B(c^{x+t}-c^{x})/ln[c])} = e^{-At} g^{c^{x} (c^{t}-1)} </math>
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| where <math>g=e^{-B/ln[c]}.</math>
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| == See also ==
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| * [[Failure rate]]
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| * [[Hazard function]]
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| * [[Actuarial present value]]
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| * [[Actuarial science]]
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| * [[Reliability theory]]
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| == References ==
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| {{Reflist}}
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| ==External links==
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| *http://www.fenews.com/fen46/topics_act_analysis/topics-act-analysis.htm
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| [[Category:Actuarial science]]
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| [[Category:Demography]]
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