## Birnbaum-Saunders Distribution

### Definition

The Birnbaum-Saunders distribution has the density function

`$\frac{1}{\sqrt{2\pi }}\mathrm{exp}\left\{-\frac{{\left(\sqrt{x}{\beta }}-\sqrt{\beta }{x}}\right)}^{2}}{2{\gamma }^{2}}\right\}\left(\frac{\left(\sqrt{x}{\beta }}+\sqrt{\beta }{x}}\right)}{2\gamma x}\right)$`

with scale parameter β > 0 and shape parameter γ > 0, for x > 0.

If x has a Birnbaum-Saunders distribution with parameters β and γ, then

`$\frac{\left(\sqrt{x}{\beta }}-\sqrt{\beta }{x}}\right)}{\gamma }$`

has a standard normal distribution.

### Background

The Birnbaum-Saunders distribution was originally proposed as a lifetime model for materials subject to cyclic patterns of stress and strain, where the ultimate failure of the material comes from the growth of a prominent flaw. In materials science, Miner's Rule suggests that the damage occurring after n cycles, at a stress level with an expected lifetime of N cycles, is proportional to n / N. Whenever Miner's Rule applies, the Birnbaum-Saunders model is a reasonable choice for a lifetime distribution model.

### Parameters

To estimate distribution parameters, us `mle` or the Distribution Fitter app.