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Birnbaum-Saunders (Fatigue Life) Distribution

NIST/SEMATECH Section 1.3.6.6.10 Birnbaum-Saunders (Fatigue Life) Distribution

PDF

BS(1, 1) PDF 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 0 0.2 0.4 0.6 f(x)

CDF

BS(1, 1) CDF 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 0 0.2 0.4 0.6 0.8 1 F(x)

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CDF

Formulas

Probability Density Function

f(x)=x/β+β/x2αx ϕ ⁣(x/β−β/xα),x>0f(x) = \frac{\sqrt{x/\beta} + \sqrt{\beta/x}}{2\alpha x}\,\phi\!\left(\frac{\sqrt{x/\beta} - \sqrt{\beta/x}}{\alpha}\right), \quad x > 0

Cumulative Distribution Function

F(x)=Φ ⁣(x/β−β/xα)F(x) = \Phi\!\left(\frac{\sqrt{x/\beta} - \sqrt{\beta/x}}{\alpha}\right)

Properties

Mean

β(1+α22)\beta\left(1 + \frac{\alpha^2}{2}\right)

Variance

β2α2(1+5α24)\beta^2 \alpha^2\left(1 + \frac{5\alpha^2}{4}\right)

Overview

The Birnbaum-Saunders distribution models fatigue life of materials subject to cyclic stress, with shape α\alpha and scale β\beta. It is derived from a physical crack-growth model and is widely used in reliability engineering.

Related Distributions

Lognormal Distribution Weibull Distribution Gamma Distribution