Understanding the Weibull Law

The Weibull distribution describes waiting times and life spans for systems exhibiting a power-law failure rate. It is parameterized by a shape value $k$ and a scale parameter $\lambda$. Its probability density is $f\left(x\right)=\frac{k}{\lambda }{\frac{x}{\lambda }}^{k-1}{e}^{-{\frac{x}{\lambda }}^k}$ for $x\ge 0$. Varying $k$ produces a wide family of curves, from exponential decay at $k=1$ to bell-shaped forms around $k=3$ and beyond.

The cumulative distribution, giving the probability that a random variable does not exceed a threshold, is $F\left(x\right)=1-e^{-{\frac{x}{\lambda }}^k}$. Reliability engineers use these formulas to model component failure and plan maintenance schedules. Because the Weibull law covers sub-exponential and super-exponential behaviors depending on $k$, it provides flexibility beyond the simpler exponential distribution.

When $k$ lies below one, the hazard rate decreases with time, representing scenarios like early-life failures that become less likely as the device proves itself. For $k>1$, the hazard increases, modeling wear-out processes such as fatigue. By adjusting $\lambda$, we rescale the distribution to represent typical life expectancy. In fields as diverse as weather forecasting, materials science, and insurance, the Weibull distribution helps quantify risk over time.

This calculator lets you plug in shape and scale values along with a measurement point $x$. Upon submission, it returns the PDF and CDF at that point. The implementation uses straightforward math.js expressions. Although seemingly simple, understanding how $k$ and $\lambda$ interact fosters intuition for reliability modeling and probability theory.

Try experimenting with values: set $k=2$ and $\lambda =5$, then vary $x$ to see how the cumulative probability rises sharply after the scale threshold. With a smaller shape parameter, the tail becomes heavier, implying occasional very large values. These insights can aid decisions about replacement schedules or risk mitigation strategies.

Behind the scenes, the distribution’s moments can be computed in closed form. The expected value is $\lambda \Gamma \left(1+\frac{1}{k}\right)$. Likewise, the variance involves $\Gamma \left(1+\frac{2}{k}\right)$. Such formulas reveal the interplay between shape and spread.