Weibull distribution

The default distribution of reliability engineering. A single shape parameter $k$ lets it model everything from “failures keep happening forever” ( $k < 1$ ) to “everything is fine until it suddenly isn’t” ( $k ≫ 1$ ).

f (x; λ, k) = \frac{k}{λ} (\frac{x}{λ})^{k - 1} e^{- (x / λ)^{k}}

PDF (mean 0.90, mode 0.48) hazard rate

h (x) = \frac{k}{λ} (\frac{x}{λ})^{k - 1}

λ (scale) 1.0 k (shape) 1.5 n (samples) 1000

PDF

f (x; λ, k) = \frac{k}{λ} (\frac{x}{λ})^{k - 1} e^{- (x / λ)^{k}}

. Hazard is decreasing for

k < 1

(infant mortality), constant at

k = 1

(exponential), rising for

k > 1

(wear-out).

The bathtub curve

The defining feature of the Weibull is how its hazard rate — the instantaneous probability of failure given survival so far — depends on shape:

h (x) = \frac{k}{λ} (\frac{x}{λ})^{k - 1}

k < 1: decreasing hazard. Components most likely to fail have already failed. Infant mortality. Think electronics that either die immediately or last forever.
k = 1: constant hazard. Memoryless. Pure exponential. The component has no memory of how long it’s been running.
k > 1: increasing hazard. The longer it’s been running, the more likely it is to fail. Wear-out. Bearings, lightbulbs, hard drives near end of life.

The dashed red curve in the demo shows the hazard rate; the lime curve is the PDF.

Why not just exponential?

The exponential distribution has a constant hazard rate — useful, but limiting. Weibull generalizes it with essentially the same effort: one extra parameter, one analytic CDF, one trivial inverse-CDF sampler. That’s why Weibull dominates survival analysis, wind-speed modelling, and materials fatigue.

E [X] = λ Γ (1 + \frac{1}{k})