Slides - Åbo Akademi

Appendix: Failure Rate Proof
Proof:
1
λ(t) = lim
∆t→0 ∆t · P {A[t,t+∆t] S
| ¬A [0,t]
S
} =
lim
∆t→0
lim
∆t→0
1
∆t · P {A[t,t+∆t] S
P {¬A [0,t]
∩ ¬A [0,t]
S
}
S
}
1
∆t · P {A[0,t+∆t] S
} − P {A [0,t]
R(t)
lim
∆t→0
S
}
=
=
R(t) − R(t + ∆t)
∆tR(t)
= f (t)
R(t) ,
where
– A [x,y]
S
̂= S failed over time [x, y]
– ¬A [x,y]
S
̂= S did not fail over time [x, y]