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The Birnbaum-Saunders distribution has the density function
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
has a standard normal distribution.
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.
To estimate distribution parameters, us mle or the Distribution Fitting app.