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Weibull distribution

What it describes

Weibull is a principal family for lifetimes and strength. Its shape parameter allows failure hazard to decrease, remain constant, or increase with age.

The variable is positive. alpha controls shape and beta scale in Phitter’s convention; check whether another text swaps those roles.

History and names

Waloddi Weibull presented a widely applicable distribution in 1951 and demonstrated it for material strength. Reliability adopted it because of its flexible hazard.

How it connects to other distributions

Shape 1 gives Exponential; shape 2 connects to Rayleigh and Gamma transforms. In extremes, Weibull is the type-III regime of GEV for maxima with a finite upper endpoint.

Where it appears

  • reliability of components, materials, and machines
  • fracture strength, survival, and warranty analysis

Fitting cautions

A stable shape parameter in one group does not imply the same mechanism in another; compare hazards and censoring before scales.

Three possible ages for a machine

Its shape parameter turns Weibull into three ageing stories. Below one, hazard decreases and may represent a population that loses its fragile units early. At one, it recovers the constant Exponential rate. Above one, hazard rises with wear or accumulated damage.

Waloddi Weibull argued for the function’s wide applicability in 1951, but its usefulness also comes from the weakest-link model for material strength. A Weibull plot makes the appropriate relationship roughly linear when the family fits. Systematic curvature can reveal mixtures, competing failure modes, or a missing threshold. Censoring components that still operate is not a minor inconvenience: it belongs in the likelihood for the experiment.

Decision guide

A good candidate when: positive lifetimes have a monotone hazard—decreasing, constant, or increasing according to shape.

Compare it with: Exponential for constant hazard, and Gamma or Lognormal for other dynamics. Include censoring and inspect hazard, not only the histogram of observed failures.

References

  • SciPy reference: scipy.stats.weibull_min — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1994). Continuous Univariate Distributions, 2nd ed., Vol. 1. Wiley.
  • Weibull, W. (1951). A statistical distribution function of wide applicability. Journal of Applied Mechanics, 18, 293–297.
  • Lawless, J. F. (2003). Statistical Models and Methods for Lifetime Data, 2nd ed. Wiley.

Weibull Distribution: equations and calculator

Distribution defintion

X∼Weibull(α,β)X\sim\mathrm{Weibull}\left(\alpha,\beta\right)

Distribution domain

x∈[0,∞)x\in [0,\infty)

Parameters domain and parameters constraints

α∈R+,β∈R+\alpha\in\mathbb{R}^{+},\beta\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=1−e−(x/β)αF_{X}\left(x\right)=1-e^{-(x/\beta)^\alpha}

Probability density function

fX(x)=αβ(xβ)α−1e−(x/β)αf_{X}\left(x\right)=\frac{\alpha}{\beta}\left(\frac{x}{\beta}\right)^{\alpha-1}e^{-(x/\beta)^\alpha}

Percent point function/Sample

FX−1(u)=β(−ln⁡(1−u))1/αF^{-1}_{X}\left(u\right)=\beta(-\ln(1-u))^{1/\alpha}

Non-central parametric moments

μk′=E[Xk]=∫0∞xkfX(x)dx=βαΓ(1+kα)\mu'_{k}=E[X^k]=\int_{0}^{\infty }x^{k}f_{X}\left(x\right)dx=\beta^\alpha \Gamma\left(1+\frac{k}{\alpha}\right)

Parametric mean

Mean(X)=μ1′=β⋅Γ(1+1/α)\mathrm{Mean}(X)=\mu'_{1}=\beta\cdot\Gamma(1+1/\alpha)

Parametric variance

Variance(X)=μ2′−μ1′2=β2[Γ(1+2/α)−(Γ(1+1/α))2]\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\beta^2\left[\Gamma\left(1+2/\alpha\right)-\left(\Gamma\left(1+1/\alpha\right)\right)^2\right]

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}

Parametric median

Median(X)=β(ln⁡(2))1/α\mathrm{Median}(X)=\beta(\ln(2))^{1/\alpha}

Parametric mode

Mode(X)={β(α−1α)1/αif α>10if α≤1\mathrm{Mode}(X)=\left\{\begin{array}{cl} \beta\left(\frac{\alpha-1}{\alpha}\right)^{1/\alpha} & \text{if }\alpha>1\\ 0 & \text{if } \alpha\leq 1 \end{array} \right.

Additional information and definitions

β:Scale parameter\beta:\text{Scale parameter}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Γ(x):Gamma function\Gamma\left(x\right):\text{Gamma function}