The Birnbaum-Saunders distribution represents the distribution of lifetimes for components under certain wear conditions. This unimodal distribution was originally formulated to model failures due to cracks, it is an example of a fatigue life distribution.
The Probability Density Function (PDF) for the Birnbaum-Saunders distribution is 
Where α is the shape parameter and β is the scale parameter. The distribution is a mixture of an inverse Gaussian distribution and a reciprocal inverse Gaussian distribution . Shapes of the distribution vary from highly skewed with long tails to almost symmetric as values for β increase .
There are several different variations on the formula in the literature. For example, Desmond  rewrites the formula as 
This formula relaxes some of the assumptions of Birnbaum and Saunders formula, strengthening the physical justification for using the distribution . Rieck and Nedelman  presented a log-linear model for the distribution, applicable to accelerated life-testing. If a data set that is thought to be Birnbaum–Saunders distributed, the parameters’ values are best estimated by maximum likelihood , although a Jackknife estimator is a possibility .
Properties of the Birnbaum-Saunders Distribution
 Birnbaum, Z. W., and Saunders, S. C. (1969). A new family of life distributions, Journal of Applied Probability, 6, 319-327.
 Engineering Statistics Handbook. Fatigue Life (Birnbaum-Saunders). Online: https://www.itl.nist.gov/div898/handbook/apr/section1/apr166.htm
 Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.
 Desmond, A. F. (1986). On relationship between two fatigue-life models, IEEE Transactions on Reliability, 35, 167-1 69.
 Rieck, J. R., and Nedelman, J. (1991). A log-linear model for the Birnbaum-Saunders distribution, Technometrics, 33, 51-60.
 Birnbaum, Z. W., and Saunders, S. C. (1969). Estimation for a family of life distributions with applications to fatigue, Journal of Applied Probability, 6, 328-347.
 Ahmad, I. A. (1988). Jackknife estimation for a family of life distributions, Journal of Statistical Computation and Simulation, 29, 211-223.
Stephanie Glen. "Birnbaum-Saunders Distribution" From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/birnbaum-saunders-distribution-2/
Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!
Comments? Need to post a correction? Please post a comment on our Facebook page.