Weibull Distribution PDF⚠ unverified

Understanding the Parameters

• Shape parameter $\beta$ — the heart of the Weibull. It determines the failure-rate behaviour:
• $\beta < 1$: decreasing failure rate — infant mortality (early-life defects being weeded out).

• $\beta = 1$: constant failure rate — random failures; the Weibull reduces exactly to the exponential distribution with $\lambda = 1/\eta$.

• $\beta > 1$: increasing failure rate — wear-out (aging, fatigue, corrosion). $\beta = 2$ is the Rayleigh distribution (linearly rising hazard); $\beta \approx 3.4$ approximates the normal. This single knob lets one distribution span the whole bathtub.

• Scale parameter $\eta$ — the characteristic life: the value by which $63.2\%$ of the population has failed (since $f$ and its CDF give $F(\eta) = 1 - e^{-1} \approx 0.632$, independent of $\beta$). It stretches the distribution along the $x$-axis, setting the timescale of failures.

• The value $x$ — usually a time, cycle count, or life. The density peaks near $\eta$ (for $\beta > 1$) and its shape shifts dramatically with $\beta$.

• The output $f(x)$ — a probability density (integrate for probabilities). It connects to the reliability (Weibull Reliability $R = e^{-(x/\eta)^\beta}$), hazard rate (Weibull Failure Rate), and the life statistics Weibull Mean, Weibull Median, Weibull Mode, and B10 Life — all built from these same two parameters.

Derivation (Approaching a Proof)

The Weibull PDF is the derivative of its cumulative distribution function. The Weibull CDF — the probability of failure by time $x$ — is postulated as a stretched exponential,

$$F(x) = 1 - e^{-(x/\eta)^\beta},$$

which is $0$ at $x=0$, rises to $1$ as $x\to\infty$, and passes through $1 - e^{-1}$ at $x = \eta$ (the characteristic-life property). The PDF is its derivative:

$$f(x) = \frac{dF}{dx} = -\frac{d}{dx}e^{-(x/\eta)^\beta} = e^{-(x/\eta)^\beta}\cdot\frac{\beta}{\eta}\left(\frac{x}{\eta}\right)^{\beta-1} = \frac{\beta}{\eta}\left(\frac{x}{\eta}\right)^{\beta-1}e^{-(x/\eta)^\beta}. \qquad\blacksquare$$

using the chain rule with $\frac{d}{dx}(x/\eta)^\beta = \frac{\beta}{\eta}(x/\eta)^{\beta-1}$. The corresponding hazard rate $h(x) = f/(1-F) = \frac{\beta}{\eta}(x/\eta)^{\beta-1}$ (Weibull Failure Rate) makes the $\beta$-controlled failure-rate behaviour explicit: a power law in $x$ with exponent $\beta - 1$, decreasing for $\beta<1$, flat for $\beta=1$, increasing for $\beta>1$.

Dimensional check. With $x$ and $\eta$ in the same units, $(x/\eta)$ is dimensionless; the prefactor $\beta/\eta$ carries "per unit $x$", so $f(x)$ is a density and $\int f\,dx = 1$. $\checkmark$

History and Development

• Waloddi Weibull, 1951. The Swedish engineer Waloddi Weibull popularised the distribution in a landmark 1951 paper, "A Statistical Distribution Function of Wide Applicability," demonstrating its versatility for material strength, fatigue life, and particle sizes. (The form was known earlier to Fréchet and Rosin–Rammler.)

• Extreme-value roots. The Weibull is one of the three extreme-value distributions — the limiting distribution of the minimum of many samples, which is exactly why it models "weakest-link" failures where a component fails at its weakest flaw.

• The reliability standard. Weibull analysis (plotting life data on Weibull probability paper to extract $\beta$ and $\eta$) became the dominant method in reliability engineering, from ball bearings (B10 Life) to electronics, because the shape parameter directly diagnoses the failure mode (infant mortality vs random vs wear-out).

Related Concepts: Weibull Reliability, Weibull Failure Rate, Weibull Mean, Weibull Median, Weibull Mode, B10 Life, Bathtub Curve, Normal Distribution PDF, Sample Weibull

Notes: Registry calculator weibull-pdf (unverified). Weibull PDF — correct as shipped. $\beta$ sets the failure-rate regime ($<1$ infant, $=1$ random $\Rightarrow$ exponential, $>1$ wear-out); $\eta$ is the characteristic life ($63.2\%$ failed). Registry labels $x$, $\beta$, $\eta$ dimensionless (life is a time). One extreme-value (weakest-link) distribution. Returns $0$ for non-positive inputs.