 Reliability theory

Reliability theory uses probability theory and statistical inference to describe whether a component or system continues to perform its specified function. For a nonnegative failure time $T$, its reliability at time $t$ is the survival function $R(t)=\mathbb{P}(T>t)$. If $T$ has probability density function $f$, the hazard function $h(t)=f(t)/R(t)$ describes the instantaneous failure rate conditional on survival, where $R(t)>0$.

System structure matters. For independent components with reliabilities $R_i(t)$, a series reliability system fails at the first component failure, so $R_{\mathrm{series}}(t)={\prod }_i{R}_i(t)$. A parallel reliability system works while at least one component works, so $R_{\mathrm{parallel}}(t)=1-{\prod }_i(1-R_i(t))$. Two independent components each having reliability $0.9$ therefore give $0.81$ in series and $0.99$ in parallel. These product formulas depend on independence; a common cause of failure can invalidate them.^[ref]