Slides - Åbo Akademi
Slides - Åbo Akademi
Slides - Åbo Akademi
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
Appendix: Failure Rate Proof<br />
Proof:<br />
1<br />
λ(t) = lim<br />
∆t→0 ∆t · P {A[t,t+∆t] S<br />
| ¬A [0,t]<br />
S<br />
} =<br />
lim<br />
∆t→0<br />
lim<br />
∆t→0<br />
1<br />
∆t · P {A[t,t+∆t] S<br />
P {¬A [0,t]<br />
∩ ¬A [0,t]<br />
S<br />
}<br />
S<br />
}<br />
1<br />
∆t · P {A[0,t+∆t] S<br />
} − P {A [0,t]<br />
R(t)<br />
lim<br />
∆t→0<br />
S<br />
}<br />
=<br />
=<br />
R(t) − R(t + ∆t)<br />
∆tR(t)<br />
= f (t)<br />
R(t) ,<br />
where<br />
– A [x,y]<br />
S<br />
̂= S failed over time [x, y]<br />
– ¬A [x,y]<br />
S<br />
̂= S did not fail over time [x, y]