Density asks when do units fail? Hazard asks a sharper question: of the units still working right now, what fraction breaks in the next hour? Drag the line and watch all three curves move together — one population, three ways of interrogating it.
h(t) = f(t) / R(t). The density spreads failures across everyone who will ever fail; the hazard rate charges them only to the units still alive. That denominator keeps shrinking, so hazard can climb steeply while the density is already falling away.
Hazards from independent failure modes simply add. Manufacturing defects (falling), random shocks (flat) and wear-out (rising) sum to the familiar tub — which is why burn-in buys you the left wall and preventive replacement caps the right one.
Conditional reliability R(t+s | t) = R(t+s) / R(t) is the chance a unit that already survived to t completes one more mission of length s. Constant hazard means that answer never changes with age — a used part is as good as new.
| Life | Reliability R | Age (h) | Hazard there (per 1000 h) | Cum. hazard H |
|---|