# 失效率

## 离散定义下的失效率

${\displaystyle \lambda (t)={\frac {f(t)}{R(t)}}}$ ，其中${\displaystyle f(t)}$ 為（第一次）失效發生時間的分佈（失效密度函數），而${\displaystyle R(t)=1-F(t)}$ .
${\displaystyle \lambda (t)={\frac {R(t_{1})-R(t_{2})}{(t_{2}-t_{1})\cdot R(t_{1})}}={\frac {R(t)-R(t+\triangle t)}{\triangle t\cdot R(t)}}\!}$

## 連續定義下的失效率

${\displaystyle h(t)=\lim _{\Delta t\to 0}{\frac {R(t)-R(t+\Delta t)}{\Delta t\cdot R(t)}}.}$

${\displaystyle \operatorname {Pr} (T\leq t)=F(t)=1-R(t),\quad t\geq 0.\!}$

${\displaystyle F(t)=\int _{0}^{t}f(\tau )\,d\tau .\!}$

${\displaystyle h(t)={\frac {f(t)}{1-F(t)}}={\frac {f(t)}{R(t)}}.}$

${\displaystyle F(t)=\int _{0}^{t}\lambda e^{-\lambda \tau }\,d\tau =1-e^{-\lambda t},\!}$

${\displaystyle h(t)={\frac {f(t)}{R(t)}}={\frac {\lambda e^{-\lambda t}}{e^{-\lambda t}}}=\lambda .}$

## 失效率遞減

DFR的隨機變數混合後仍為DFR[4]，而指数分布的隨機變數混合後也是為DFR[5]

## 失效率資料

### 舉例

${\displaystyle {\frac {6{\text{ failures}}}{7502{\text{ hours}}}}=0.0007998{\frac {\text{failures}}{\text{hour}}}=799.8\times 10^{-6}{\frac {\text{failures}}{\text{hour}}},}$

## 參考資料

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7. ^ On Time, Reliability, and Spacecraft. Wiley-Blackwell. : 1–8 [2018-04-02]. doi:10.1002/9781119994077.ch1 （英语）.
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