Weibull Mean⚠ unverified

Understanding the Parameters

• Scale parameter $\eta$ — the characteristic life, setting the timescale. The mean is $\eta$ times a shape-dependent factor, so $\bar t$ scales linearly with $\eta$: double the characteristic life and the mean life doubles.

• Shape parameter $\beta$ — enters through $\Gamma(1 + 1/\beta)$, the correction for the distribution's shape. The factor is not intuitive but has clean landmarks: at $\beta = 1$, $\Gamma(2) = 1$, so $\bar t = \eta$ (the exponential mean equals the characteristic life). For $\beta > 1$ the factor dips slightly below $1$ (e.g. $\beta = 2$: $\Gamma(1.5) = \sqrt\pi/2 \approx 0.886$, so $\bar t \approx 0.886\,\eta$), reflecting the left-shift of the mean relative to $\eta$ as the distribution becomes more symmetric.

• The output $\bar t$ — the true average lifetime, the first moment of the distribution. It is the number to use for total expected life, spares provisioning over a population, and life-cycle cost — anything that depends on the average, not a percentile.

• Mean vs characteristic life. A common error is to equate the mean life with $\eta$. They coincide only at $\beta = 1$; otherwise the gamma factor separates them. Likewise the mean differs from the median (50th percentile) and mode (most likely) except in special cases — for a skewed distribution these three measures genuinely differ.

Derivation (Approaching a Proof)

The mean is the first moment of the Weibull density, $\bar t = \int_0^\infty t\,f(t)\,dt$ with $f(t) = \frac{\beta}{\eta}(t/\eta)^{\beta-1}e^{-(t/\eta)^\beta}$ (Weibull Distribution PDF). Substitute $u = (t/\eta)^\beta$, so $t = \eta\,u^{1/\beta}$ and $du = \frac{\beta}{\eta}(t/\eta)^{\beta-1}\,dt$ — which is exactly $f(t)\,dt/e^{-u}\cdot e^{-u}$... more directly, the substitution turns the integral into

$$\bar t = \int_0^\infty t\,f(t)\,dt = \eta\int_0^\infty u^{1/\beta}\,e^{-u}\,du.$$

The remaining integral is the definition of the gamma function, $\Gamma(z) = \int_0^\infty u^{z-1}e^{-u}\,du$, with $z - 1 = 1/\beta$, i.e. $z = 1 + 1/\beta$. Therefore

$$\bar t = \eta\,\Gamma\!\left(1 + \frac{1}{\beta}\right). \qquad\blacksquare$$

The gamma function is the continuous generalisation of the factorial ($\Gamma(n) = (n-1)!$ for integers), and its appearance here is why the Weibull mean carries a non-elementary factor. At $\beta = 1$, $\Gamma(2) = 1! = 1$, recovering the exponential mean $\bar t = \eta = 1/\lambda$ (Mtbf From Failure Rate).

Dimensional check. $\Gamma(1 + 1/\beta)$ is a dimensionless number; $\eta$ carries the time units, so $\bar t$ is a time. $\checkmark$

History and Development

• The gamma function. Introduced by Euler in the 1720s as the continuous extension of the factorial, the gamma function pervades probability; its appearance in the Weibull mean is a clean example of why "average life" is subtler than it looks for skewed distributions.

• Mean vs percentile life. Reliability practice increasingly prefers percentile life metrics (B10 Life, Weibull Median) over the mean, because for wear-out distributions the mean can be a poor guide to when failures actually start — a lesson from bearing and fatigue life data.

• Population planning. The mean life is nonetheless essential for fleet-level decisions: expected total failures, spares demand, and warranty reserves over a population all depend on the average, making $\eta\,\Gamma(1+1/\beta)$ a routine reliability computation.

Related Concepts: Weibull Median, Weibull Mode, B10 Life, Weibull Reliability, Weibull Distribution PDF, Weibull Failure Rate, Mtbf From Failure Rate

Notes: Registry calculator weibull-mean (unverified). $\bar t = \eta\,\Gamma(1+1/\beta)$ — correct as shipped; not equal to $\eta$ except at $\beta=1$ (where $\Gamma(2)=1$). The gamma factor corrects for skew. Distinct from the median and mode. Registry labels $\eta$ dimensionless (is a time).