Weibull Reliability⚠ unverified

Understanding the Parameters

• Shape parameter $\beta$ — sets the aging behaviour: $\beta < 1$ = decreasing hazard (infant mortality), $\beta = 1$ = constant hazard (reduces to exponential, $R = e^{-t/\eta}$), $\beta > 1$ = increasing hazard (wear-out). The steepness of the reliability drop near the characteristic life grows with $\beta$: high-$\beta$ items survive reliably then fail in a narrow band (a sharp "wear-out cliff"), while low-$\beta$ items fail spread across a wide range.

• Scale parameter $\eta$ (characteristic life) — the time at which $R = e^{-1} \approx 0.368$, i.e. $63.2\%$ have failed — regardless of $\beta$ (set $t = \eta$: $R = e^{-1}$). It stretches the survival curve along the time axis, setting the overall timescale of failures.

• Time $t$ — where survival is evaluated. $R$ starts at $1$ and decays; the shape of the decay (gradual vs cliff-like) is $\beta$, the timescale is $\eta$.

• The output $R$ — survival probability. Its complement $1 - R$ is the fraction failed by $t$; the specific fractions define life metrics like B10 Life ($1-R = 0.10$) and the median ($1-R = 0.50$).

Derivation (Approaching a Proof)

Reliability follows from integrating the hazard rate. The general relation between survival and hazard is $R(t) = \exp\!\big(-\int_0^t h(\tau)\,d\tau\big)$ (from $dR/R = -h\,dt$, Exponential Reliability). The Weibull model postulates a power-law hazard (Weibull Failure Rate),

$$h(t) = \frac{\beta}{\eta}\left(\frac{t}{\eta}\right)^{\beta-1}.$$

Integrating this cumulative hazard:

$$\int_0^t h(\tau)\,d\tau = \int_0^t \frac{\beta}{\eta}\left(\frac{\tau}{\eta}\right)^{\beta-1}d\tau = \left(\frac{t}{\eta}\right)^{\beta},$$

since the integrand is exactly the derivative of $(\tau/\eta)^\beta$. Therefore

$$R(t) = \exp\!\left[-\left(\frac{t}{\eta}\right)^{\beta}\right]. \qquad\blacksquare$$

The bracket $(t/\eta)^\beta$ is the cumulative hazard, and its power-law form is what gives the Weibull its adaptability. Setting $\beta = 1$ recovers the exponential $R = e^{-t/\eta}$; $\beta = 2$ gives the Rayleigh (linearly rising hazard). The complementary CDF $F(t) = 1 - R$ is the Weibull CDF whose derivative is the PDF.

Dimensional check. With $t$, $\eta$ in the same time units, $(t/\eta)^\beta$ is dimensionless, so $R$ is a dimensionless probability. $\checkmark$

History and Development

• Weibull's flexibility. Waloddi Weibull's 1951 distribution (Weibull Distribution PDF) became the reliability standard precisely because its shape parameter lets one model span failure modes that the exponential cannot — the great practical advance over constant-hazard analysis.

• Weibull plotting. Life data are traditionally plotted on Weibull probability paper (log time vs log-log of $1/R$), on which Weibull data fall on a straight line whose slope is $\beta$ and whose intercept gives $\eta$ — a graphical estimator that made the method accessible before computers.

• Diagnosing failure modes. Because $\beta$ distinguishes infant mortality, random, and wear-out failures, fitting Weibull to field data does more than predict reliability — it diagnoses why things fail, guiding burn-in (Bathtub Curve), maintenance intervals, and design fixes.

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

Notes: Registry calculator weibull-reliability (unverified). $R = e^{-(t/\eta)^\beta}$ — correct as shipped. $\beta$ sets aging (<1 infant, =1 exponential, >1 wear-out); $\eta$ is the characteristic life ($R(\eta)=e^{-1}$, $63.2\%$ failed). Cumulative hazard $(t/\eta)^\beta$. Generalises Exponential Reliability.