Weibull Mode⚠ unverified

Understanding the Parameters

• Shape parameter $\beta$ — determines whether a mode even exists. For $\beta \le 1$ the density is monotonically decreasing (highest at $t = 0$), so the mode is $0$: for infant-mortality and random-failure items, the most likely failure time is immediately — there is no interior peak. Only for wear-out ($\beta > 1$) does the density rise then fall, producing an interior mode where failures bunch up.

• Scale parameter $\eta$ — sets the timescale; the mode is $\eta$ times the shape factor $((\beta-1)/\beta)^{1/\beta}$, which is always $< 1$, so the mode is less than the characteristic life. As $\beta \to \infty$ the factor $\to 1$ and the mode approaches $\eta$ (the distribution becomes a spike at $\eta$).

• The output $t_{mode}$ — the single most probable failure time, the density's peak. It is the smallest of the three central measures for a wear-out Weibull: $t_{mode} < t_{50} < \bar t$ (mode $<$ median $<$ mean), the ordering characteristic of a right-skewed distribution.

• Interpretation. The mode answers "if this part fails, when is it most likely to happen?" — distinct from the average (mean) and the half-failed point (median). For maintenance it flags the age of peak failure density.

Derivation (Approaching a Proof)

The mode is where the probability density is maximised, found by setting its derivative to zero. Take the Weibull PDF $f(t) = \frac{\beta}{\eta}(t/\eta)^{\beta-1}e^{-(t/\eta)^\beta}$ (Weibull Distribution PDF). It is easier to maximise $\ln f$ (the log is monotonic, so it shares the maximum):

$$\ln f = \text{const} + (\beta-1)\ln t - \left(\frac{t}{\eta}\right)^\beta.$$

Differentiate with respect to $t$ and set to zero:

$$\frac{d\ln f}{dt} = \frac{\beta-1}{t} - \frac{\beta}{\eta}\left(\frac{t}{\eta}\right)^{\beta-1} = 0.$$

Multiply through by $t$ and rearrange: $\frac{\beta}{\eta^\beta}t^\beta = \beta - 1$, so $(t/\eta)^\beta = (\beta-1)/\beta$, giving

$$t_{mode} = \eta\left(\frac{\beta-1}{\beta}\right)^{1/\beta}. \qquad\blacksquare$$

The condition requires $\beta > 1$ for a positive solution; for $\beta \le 1$ the derivative is negative everywhere (density decreasing), so the maximum sits at the boundary $t = 0$ — hence the piecewise definition.

Dimensional check. $((\beta-1)/\beta)^{1/\beta}$ is a dimensionless number; $\eta$ carries the time units, so $t_{mode}$ is a time. $\checkmark$

History and Development

• The three central measures. Mode, median, and mean are the classical measures of central tendency; for the symmetric normal they coincide, but for a skewed distribution like the Weibull they separate in a fixed order (mode $<$ median $<$ mean for right skew) — a textbook illustration of why "average" is ambiguous.

• Wear-out signature. The existence of an interior mode only for $\beta > 1$ is another face of the failure-mode diagnosis: wear-out concentrates failures around a characteristic age, while infant-mortality and random failures do not, peaking at $t = 0$.

• Maintenance planning. The mode marks the age of maximum failure density, a natural reference point for scheduling preventive replacement shortly before the failure cluster — complementing percentile-based (B10 Life) and hazard-based (Weibull Failure Rate) maintenance criteria.

Related Concepts: Weibull Mean, Weibull Median, B10 Life, Weibull Distribution PDF, Weibull Reliability, Weibull Failure Rate

Notes: Registry calculator weibull-mode (unverified). $t_{mode} = \eta((\beta-1)/\beta)^{1/\beta}$ for $\beta>1$, else $0$ — correct as shipped. No interior mode for $\beta\le1$ (density peaks at $t=0$). Ordering mode $<$ median $<$ mean (right skew). Registry labels $\eta$ dimensionless (is a time).