< Probability Distributions List

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.

**Contents:**

## Birnbaum-Saunders Distribution properties

The Probability Density Function (PDF) for the Birnbaum-Saunders distribution is [1]:

Where α is the shape parameter and β is the scale parameter.

The distribution is a mixture distribution of an inverse Gaussian distribution and a reciprocal inverse Gaussian distribution [2]. Shapes of the distribution vary from highly skewed with long tails to almost symmetric as values for β increase [2].

There are several different variations on the formula in the literature. For example, Desmond [3] rewrites the formula as [4]:

This formula relaxes some of the assumptions of Birnbaum and Saunders formula, strengthening the physical justification for using the distribution [5].

Rieck and Nedelman [5] 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 [6], although a Jackknife estimator is a possibility [7].

## Properties of the Birnbaum-Saunders Distribution

## Other types of fatigue life distribution

Various fatigue life distributions exist, each with their own pros and cons. While the Birnbaum-Saunders distribution is commonly known as “the”

fatigue life distribution [1], there are some other types commonly used to model fatigue life data. They include:

- Normal distribution: Often used for modeling fatigue life data, the normal distribution is symmetric and relatively easy to comprehend and apply. It enables predictions about a product’s fatigue life. However, it may be inaccurate for specific types of fatigue life data, such as highly skewed data or data with a long tail.
- Log-normal distribution: This right-skewed distribution is frequently used to model highly skewed fatigue life data. While it offers greater accuracy than the normal distribution for this kind of data, it can be more challenging to understand and use.
- Weibull distribution: This is a versatile option for modeling a wide range of fatigue life data. It generally provides better accuracy than the normal distribution for most fatigue life data types and is relatively simple to comprehend and use.
- Gompertz-Makeham distribution: This two-parameter distribution is commonly used for modeling fatigue life data characterized by a decreasing failure rate. Although it offers higher accuracy than the normal distribution for such data, it can be more difficult to understand and use.

The selection of a fatigue life distribution depends on the data being modeled and the required accuracy level. Generally, the normal distribution is suitable for non-highly skewed fatigue life data, while the log-normal or Weibull distribution is appropriate for highly skewed data. The Gompertz-Makeham distribution is an ideal choice for fatigue life data characterized by a decreasing failure rate.

Some key features of the different types of fatigue life distribution:

Distribution | Shape | Symmetry | Mean | Variance |
---|---|---|---|---|

Normal distribution | Bell-shaped | Symmetric | μ | σ^{2} |

Lognormal distribution | Right-skew | Asymmetrical | μ+σln(z) | (ln(σ^{2}z))^{2} |

Weibull distribution | Right-skew | Asymmetrical | βΓ(α) | β^{2}Γ(α)^{2} |

Gompertz-Makeham distribution | Right-skew | Asymmetrical | μexp(βt) | μ^{2}β^{2}exp(2βt) |

Where:

*μ*= mean*σ*= standard deviation*α*= shape parameter*β*= scale parameter*z*= random variable- Γ = Gamma function

## Fatigue life distribution uses

Fatigue life distributions are used to estimate a product’s fatigue life, which refers to the number of cycles a product can endure before failure. This distribution helps in determining the likelihood of a product failing after a specific number of cycles.

Additionally, fatigue life distribution aids in designing products with greater resistance to fatigue failure. By understanding the product’s fatigue life distribution, engineers can develop products with extended fatigue life, thereby reducing the probability of failure.

## References

[1] Birnbaum, Z. W., and Saunders, S. C. (1969). A new family of life distributions, Journal of Applied Probability, 6, 319-327.

[2] Engineering Statistics Handbook. Fatigue Life (Birnbaum-Saunders). Online: https://www.itl.nist.gov/div898/handbook/apr/section1/apr166.htm

[3] Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.

[4] Desmond, A. F. (1986). On relationship between two fatigue-life models, IEEE Transactions on Reliability, 35, 167-1 69.

[5] Rieck, J. R., and Nedelman, J. (1991). A log-linear model for the Birnbaum-Saunders distribution, Technometrics, 33, 51-60.

[6] 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.

[7] Ahmad, I. A. (1988). Jackknife estimation for a family of life distributions, Journal of Statistical Computation and Simulation, 29, 211-223.

[8] NIST. Birnbaum-Saunders (Fatigue Life) Distribution. Retrieved June 8, 2023 from:https://www.itl.nist.gov/div898/handbook/eda/section3/eda366a.htm