Reliability theory
= 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 $R_{\rm series}(t)=\prod_i R_i(t)$. A <parallel reliability system> works while at least one component works, so $R_{\rm parallel}(t)=1-\prod_i(1-R_i(t))$. 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.<NIST and SEMATECH e-Handbook of Statistical Methods>[Sections 8.1.8.2–8.1.8.3, series and parallel system models]{ref}