Question

Derive the general equation for failure-time distribution

Answer 1

To derive the relationship expressing the failure-time density in terms of the failure-rate function. Make use of the fact that \(R(t)=1-F(t)\) and, hence, that \(F^{\prime}(t)=-R^{\prime}(t)\), we can write
\[
Z(t)=-\frac{R^{\prime}(t)}{R(t)}=-\frac{d[\ln R(t)]}{d t}
\]
Solving this differential equation for \(R(t)\), we obtain
\[
R(t)=e^{-\int_{0}^{t}} Z(x) d x
\]
and, making use of the relation \(f(t)=Z(t) \cdot R(t)\), we finally get

\[f(t)=Z(t) \cdot e^{-\int_{0}^{t} Z(x) d x}\]