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Derive the General equation for failure-rate function
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Let $f(t)$ be the probability density of the time to failure of a given component. The probability that the component will fail between times $t$ and $t+\Delta t$ is approximately $f(t) \cdot \Delta t$. Then, the probability that the component will fail on the interval from 0 to $t$ is given by
$F(t)=\int_{0}^{t} f(x) d x$
and the reliability function, expressing the probability that it survives to time $t$, is given by
$R(t)=1-F(t)$
We can then express the probability that the component will fail in the interval from $t$ to $t+\Delta t$ as $F(t+\Delta t)-F(t)$, and the conditional probability of failure in this interval, given that the component survived to time $t$, is expressed by
$\frac{F(t+\Delta t)-F(t)}{R(t)}$
Dividing by $\Delta t$, we find that the average rate of failure in the interval from $t$ to $t+\Delta t$, given that the component survived to time $t$, is
$\frac{F(t+\Delta t)-F(t)}{\Delta t} \cdot \frac{1}{R(t)}$
Taking the limit as $\Delta t \rightarrow 0$, we then get the instantaneous failure rate, or simply the failure rate or hazard rate
$Z(t)=\frac{F^{\prime}(t)}{R(t)}$
where $F^{\prime}(t)$ is the derivative of $F(t)$ with respect to $t$. Finally, observing that $f(t)=$ $F^{\prime}(t)$, we get the relation
$Z(t)=\frac{f(t)}{R(t)}=\frac{f(t)}{1-F(t)}$
by Wooden Status (4,147 points)

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