217 Appendix C Appendix: Multi-State Deterioration C.1 Relations in reliability analysis From probability theory, in an absolutely continuous univariate distribution, a random variable T has a density function fT, where fT is a non-negative Lebesgueintegrable function: P[a ⇓T ⇓b] = b a fT(t)dt (C.1) The cumulative distribution function FT(t) is: P[T ⇓t] =FT(t)= t ↗↘ fT(u)du (C.2) The survival function ST(t), mathematically equivalent to the reliability function R(t), is: P[T ⇒t] =S T(t)=R(t)=1→FT(t)= ↘ t fT(u)du (C.3) Now, let the hazard function ϖ(t) be defined as: ϖ(t)= lim ∆t⇐0 P[t ⇓T <t +∆t|T ⇒t] ∆t (C.4) When applying Bayes theorem in Eq. C.4, we get: P[t ⇓T <t +∆t|T ⇒t] = P[(t ⇓T <t +∆t) ↖(T ⇒t)] P[T ⇒t] Since T ⇒t is part of the event t ⇓T <t +∆, we simplify: P[(t ⇓T <t +∆) ↖(T ⇒t)] =P[t ⇓T <t +∆t]
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