Weibull Failure Rate⚠ unverified

Understanding the Parameters

• Shape parameter $\beta$ — through the exponent $\beta - 1$ it sets the trend of the failure rate:

• $\beta < 1$: exponent negative $\Rightarrow$ hazard decreases with time (infant mortality — weak units die early, survivors are stronger).

• $\beta = 1$: exponent zero $\Rightarrow$ hazard constant at $1/\eta$ (random failures; the exponential case).

• $\beta > 1$: exponent positive $\Rightarrow$ hazard increases with time (wear-out — aging, fatigue, corrosion). $\beta = 2$ gives a linearly rising hazard. This single parameter is the diagnostic: fit $\beta$ from field data and you learn the failure mechanism.

• Scale parameter $\eta$ — sets the timescale. At $t = \eta$ the hazard equals $\beta/\eta$ regardless of the power, anchoring the rate. Larger $\eta$ shifts the whole hazard curve to later times.

• The output $h(t)$ — the instantaneous failure rate, in failures per unit time. It differs from the cumulative average rate; it is the "right now" risk given survival to $t$. Integrating it gives the cumulative hazard $(t/\eta)^\beta$ that drives $R(t) = e^{-(t/\eta)^\beta}$.

• Registry unit labels. The registry labels $t$ and $\eta$ dimensionless; both are times, and $h$ is a rate (1/time).

Derivation (Approaching a Proof)

The hazard rate is defined as the ratio of the failure density to the survival function, $h(t) = f(t)/R(t)$ — the failure density conditioned on having survived. For the Weibull, the PDF is $f(t) = \frac{\beta}{\eta}(t/\eta)^{\beta-1}e^{-(t/\eta)^\beta}$ (Weibull Distribution PDF) and the survival is $R(t) = e^{-(t/\eta)^\beta}$ (Weibull Reliability). Dividing:

$$h(t) = \frac{f(t)}{R(t)} = \frac{\frac{\beta}{\eta}(t/\eta)^{\beta-1}e^{-(t/\eta)^\beta}}{e^{-(t/\eta)^\beta}} = \frac{\beta}{\eta}\left(\frac{t}{\eta}\right)^{\beta-1}. \qquad\blacksquare$$

The exponential survival terms cancel, leaving the clean power law. This cancellation is special to the Weibull form and is exactly why the model is so tractable: the hazard is elementary, the cumulative hazard is $(t/\eta)^\beta$, and reliability is its exponential. The ratio of hazards at two times, $h(t_2)/h(t_1) = (t_2/t_1)^{\beta-1}$, is independent of $\eta$ (Hazard Rate Ratio) — a convenient way to compare risk across ages.

Dimensional check. $(t/\eta)^{\beta-1}$ is dimensionless; the prefactor $\beta/\eta$ carries $1/\text{time}$, so $h(t)$ is a rate. $\checkmark$

History and Development

• The diagnostic curve. Plotting the hazard rate versus time is the clearest way to see a component's failure character, and the Weibull hazard's monotonic power-law shape is the building block of the composite bathtub (decreasing + constant + increasing pieces).

• Burn-in and maintenance. A decreasing hazard ($\beta < 1$) justifies burn-in (run parts long enough to weed out early failures before shipping); an increasing hazard ($\beta > 1$) justifies preventive replacement before wear-out accelerates. The sign of $\beta - 1$ dictates the maintenance strategy.

• Proportional hazards. The Weibull hazard underlies accelerated life testing and proportional-hazards models, where stress multiplies the hazard, letting short high-stress tests predict long service-life reliability.

Related Concepts: Weibull Reliability, Weibull Distribution PDF, Hazard Rate Ratio, Weibull Mean, B10 Life, Exponential Reliability, Bathtub Curve

Notes: Registry calculator weibull-failure-rate (unverified). $h(t) = \frac{\beta}{\eta}(t/\eta)^{\beta-1}$ — correct as shipped; power law with exponent $\beta-1$ (decreasing/constant/increasing hazard). $h = f/R$; integrates to the cumulative hazard $(t/\eta)^\beta$. Registry labels $t$, $\eta$ dimensionless (are times).