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)}

\]