Birnbaum-Saunders distribution

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The Birnbaum-Saunders distribution (also known as the fatigue life distribution) is used extensively in reliability applications to model failure times. There are several alternative formulations of this distribution in the literature.

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[edit] Probability density function

The general formula for the probability density function (pdf) is

f(x) = \frac{\sqrt{\frac{x-\mu}{\beta}}+\sqrt{\frac{\beta}{x-\mu}}}{2\gamma\left(x-\mu\right)}\phi\left(\frac{\sqrt{\frac{x-\mu}{\beta}}-\sqrt{\frac{\beta}{x-\mu}}}{\gamma}\right)\quad x > \mu; \gamma,\beta>0

where γ is the shape parameter, μ is the location parameter, β is the scale parameter, and \phi \, is the probability density function of the standard normal distribution.

[edit] Standard fatigue life distribution

The case where μ = 0 and β = 1 is called the standard fatigue life distribution. The pdf for the standard fatigue life distribution reduces to

f(x) = \frac{\sqrt{x}+\sqrt{\frac{1}{x}}}{2\gamma x}\phi\left(\frac{\sqrt{x}-\sqrt{\frac{1}{x}}}{\gamma}\right)\quad x > 0; \gamma >0

Since the general form of probability functions can be expressed in terms of the standard distribution, all of the subsequent formulas are given for the standard form of the function.

[edit] Cumulative distribution function

The formula for the cumulative distribution function is

F(x) = \Phi\left(\frac{\sqrt{x} - \sqrt{\frac{1}{x}}}{\gamma}\right)\quad x > 0; \gamma > 0

where Φ is the cumulative distribution function of the standard normal distribution.

[edit] Quantile function

The formula for the quantile function is

G(p) = \frac{1}{4}\left[\gamma\Phi^{-1}(p) + \sqrt{4+\left(\gamma\Phi^{-1}(p)\right)^2}\right]^2

where Φ −1 is the quantile function of the standard normal distribution.

[edit] External links

[edit] Further reading

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This article incorporates text from a public domain publication of the National Institute of Standards and Technology, a U.S. government agency.