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Weibull Distribution

NIST/SEMATECH Section 1.3.6.6.8 Weibull Distribution

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Weibull(1, 1) PDF 0 0.5 1 1.5 2 2.5 3 x 0 0.2 0.4 0.6 0.8 1 f(x)

CDF

Weibull(1, 1) CDF 0 0.5 1 1.5 2 2.5 3 x 0 0.2 0.4 0.6 0.8 1 F(x)

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CDF

Formulas

Probability Density Function

f(x)=αβ(xβ)α−1e−(x/β)α,x≥0f(x) = \frac{\alpha}{\beta}\left(\frac{x}{\beta}\right)^{\alpha-1} e^{-(x/\beta)^\alpha}, \quad x \ge 0

Cumulative Distribution Function

F(x)=1−e−(x/β)α,x≥0F(x) = 1 - e^{-(x/\beta)^\alpha}, \quad x \ge 0

Properties

Mean

β Γ ⁣(1+1α)\beta\,\Gamma\!\left(1 + \frac{1}{\alpha}\right)

Variance

β2[Γ ⁣(1+2α)−Γ2 ⁣(1+1α)]\beta^2\left[\Gamma\!\left(1+\frac{2}{\alpha}\right) - \Gamma^2\!\left(1+\frac{1}{\alpha}\right)\right]

Overview

The Weibull distribution is a versatile distribution used in reliability engineering and failure analysis. It can model increasing (α>1\alpha > 1), decreasing (α<1\alpha < 1), or constant (α=1\alpha = 1) failure rates depending on its shape parameter α\alpha.

Related Distributions

Exponential Distribution Gamma Distribution Extreme Value Type I (Gumbel) Distribution