Weibull hazard rate
Hazard rate h(t) is the intensity of failure at age t, given survival to t. Reliability work uses it to distinguish infant mortality, useful life, and wear-out. Weibull.Genie evaluates h(t) from the fitted LifeFit; this page states the 2P Weibull formula and a numerical check.
Open the Weibull.Genie workbook Open Weibull Calculator Evaluate h(t) from a fit
Formula
With density f(t) = (β/η)(t/η)β−1 exp[−(t/η)β] and reliability R(t) = exp[−(t/η)β],
h(t) = f(t) / R(t) = (β/η) (t/η)β−1
- β < 1 — decreasing hazard (DFR), infant mortality, burn-in candidates.
- β = 1 — constant hazard, exponential, h = 1/η = 1/MTTF.
- β > 1 — increasing hazard (IFR), wear-out, replacement / inspection age matters.
A single two-parameter Weibull is monotone. The decorative bathtub on the Weibull.Genie header is three example shapes overlaid, not a fitted mixture.
Worked example (β = 2, η = 1 000 h)
At t = 500 h:
h(500) = (2 / 1 000) × (500 / 1 000)1 = 0.002 × 0.5 = 0.001 / h
That is one expected failure per 1 000 surviving unit-hours at age 500 h. At t = η = 1 000 h the same model gives h = 0.002 / h — twice as steep, because β = 2 is linear wear-out in t. For β = 1, η = 1 000 h, h would stay 0.001 / h at every age.
| t (h) | R(t) | h(t) (1/h) |
|---|---|---|
| 250 | 0.939 | 0.0005 |
| 500 | 0.779 | 0.0010 |
| 1 000 | 0.368 | 0.0020 |
Fit data on the workbook, then Calculator → Failure rate. Closed-form B10 and MTTF for this same (β, η) are on the B10 and MTTF pages. A full life-data walk-through is in the tutorial.
FAQ
What is the hazard rate h(t)?
h(t) = f(t) / R(t) is the instantaneous failure rate at age t among units that have survived so far. It is not the fraction failed F(t), and it is not 1/MTTF except for the exponential.
What is the Weibull hazard formula?
h(t) = (β/η) (t/η)^{β−1} for t > 0. If β = 1 the power vanishes and h(t) = 1/η (constant). If β > 1 the hazard rises (wear-out). If β < 1 it falls (infant mortality).
How does this relate to the bathtub curve?
A bathtub is often three Weibull (or mixed) regimes in series: β < 1 early, β ≈ 1 in useful life, β > 1 in wear-out. A single 2P Weibull has a monotone hazard — it cannot draw a full bathtub by itself.
Where is h(t) in Weibull.Genie?
Weibull Calculator → Rate → Failure rate h(t). Enter the evaluation time in the same life unit as the workbook. The header bathtub artwork is decorative, not a fitted hazard plot.