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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Force of mortality ： ウィキペディア英語版 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. ==Motivation and definition== 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 has attained age (''x'') dying between ages ''x'' and ''x'' + ''Δx'', which is : where FX(x) is the cumulative 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 $\backslash mu(x)$: :$\backslash mu\backslash ,(x)=\backslash frac$ 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, the force of mortality can also be expressed variously as: :$\backslash mu\backslash ,(x)=\backslash frac=-\backslash frac=-\}\backslash ln().$ To understand conceptually how the force of mortality operates within a population, consider that the ages, ''x'', where the probability density function ''f''''X''(''x'') is zero, there is no chance of dying. Thus the force of mortality at these ages is zero. The force of mortality $\backslash mu(x)$ can be interpreted as the ''conditional'' density of failure at age ''x'', while ''f''(''x'') is the ''unconditional'' density of failure at age ''x''.〔R. Cunningham, T. Herzog, R. London (2008). ''Models for Quantifying Risk, 3rd Edition'', Actex.〕 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''. This is expressed in symbols as :$\backslash ,\backslash mu(x)S(x)\; =\; f\_X(x)$ or equivalently :$\backslash mu(x)\; =\; \backslash frac.$ 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'' :$\backslash int\_^\; -\backslash frac\; \backslash ln()\backslash ,\; dy$. By the fundamental theorem of calculus, this is simply :$\backslash ln(+\; t))\; -\; \backslash ln(),$ and taking the exponent to the base ''e'' results in :$\backslash frac\; =\; S\_x(t).$ Therefore, the survival probability of an individual of age ''x'' is written in terms of the force of mortality as :$S\_x(t)\; =\; e^.$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Force of mortality」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.