represents the BirnbaumSaunders distribution with shape parameter α and scale parameter λ.


Background & Context

  • BirnbaumSaundersDistribution[α,λ] represents a continuous statistical distribution defined over the interval and parametrized by two positive values α and λ. Here, α is known as a "shape parameter," γ is a so-called "scale parameter," and together these parameters determine various properties of the probability density function (PDF), including its height and its horizontal location in the plane. The PDF of the BirnbaumSaunders distribution is unimodal and has "thin tails" in the sense that the PDF decreases exponentially for large values . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.)
  • The BirnbaumSaunders distribution dates back to the late 1960s to the work of mathematicians Z. W. Birnbaum and S. C. Saunders and was originally proposed as a lifetime model for materials subject to cyclic patterns of stress and strain. Also known as the fatigue-life distribution, the BirnbaumSaunders distribution is still actively used to model life cycles in manufacturing. More recently, modified versions of the distribution have been used to accurately model the distribution of mineral concentration in drinking water. Elsewhere, the distribution has been used to approximate the quantile function of the inverse Gaussian distribution (InverseGaussianDistribution), to perform analyses in various areas of engineering science, and to model certain biological processes subject to rapid decline.
  • RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a BirnbaumSaunders distribution. Distributed[x,BirnbaumSaundersDistribution[α,λ]], written more concisely as xBirnbaumSaundersDistribution[α,λ], can be used to assert that a random variable x is distributed according to a BirnbaumSaunders distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
  • The probability density and cumulative distribution functions may be given using PDF[BirnbaumSaundersDistribution[α,λ],x] and CDF[BirnbaumSaundersDistribution[α,λ],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
  • DistributionFitTest can be used to test if a given dataset is consistent with a BirnbaumSaunders distribution, EstimatedDistribution to estimate a BirnbaumSaunders parametric distribution from given data, and FindDistributionParameters to fit data to a BirnbaumSaunders distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic BirnbaumSaunders distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic BirnbaumSaunders distribution.
  • TransformedDistribution can be used to represent a transformed BirnbaumSaunders distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a BirnbaumSaunders distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving beta distributions.
  • The BirnbaumSaunders distribution is related to a number of other distributions. For example, given a random variate , XBirnbaumSaundersDistribution[α,γ] if and only if YNormalDistribution[] where . Visually, the PDF of BirnbaumSaundersDistribution tends to appear "bell-shaped," thereby introducing qualitative relationships with a number of other distributions, including CauchyDistribution, StudentTDistribution, and LogisticDistribution. BirnbaumSaundersDistribution is also related to LogNormalDistribution, BetaDistribution, and JohnsonDistribution.


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Basic Examples  (4)

Probability density function:

Cumulative distribution function:

Mean and variance:


Scope  (8)

Generate a sample of pseudorandom numbers from a BirnbaumSaunders distribution:

Compare its histogram to the CDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare the density histogram of the sample with the PDF of the estimated distribution:

Skewness depends only on the shape parameter α:

The limiting value:

Kurtosis depends only on the shape parameter α:

The limiting value:

Different moments with closed forms as functions of parameters:


Closed form for symbolic order:




Closed form for symbolic order:

Hazard function:

Quantile function:

Consistent use of Quantity in parameters yields QuantityDistribution:

Find the quartiles:

Applications  (3)

The lifetime in hours of a component has a BirnbaumSaunders distribution with and per hour. Find the probability the component survives 300 hours:

Find the probability that the component is still working after 500 hours, after it has survived 300 hours:

Find the mean time to failure:

Simulate the failure times for 30 independent components like this:

The time to failure of component A follows a BirnbaumSaunders distribution with and per hour, while the failure rate of component B is 1 per hour. Find the mean time to failure for both components:

Find the probability that component A fails before component B:

Although they have the same mean lifetime, a BirnbaumSaunders distribution tends to fail early:

The lifetime of a device has a BirnbaumSaunders distribution. Find the reliability of the device:

The hazard function has horizontal asymptote :

Find the reliability of two such devices in series:

Find the reliability of two such devices in parallel:

Compare the reliability of both systems for and :

Properties & Relations  (3)

BirnbaumSaunders distribution is closed under scaling by a positive factor:

If has a BirnbaumSaunders distribution, then also has a BirnbaumSaunders distribution:

BirnbaumSaunders distribution is related to NormalDistribution:

Neat Examples  (1)

PDFs for different α values with CDF contours:

Wolfram Research (2010), BirnbaumSaundersDistribution, Wolfram Language function, (updated 2016).


Wolfram Research (2010), BirnbaumSaundersDistribution, Wolfram Language function, (updated 2016).


Wolfram Language. 2010. "BirnbaumSaundersDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016.


Wolfram Language. (2010). BirnbaumSaundersDistribution. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_birnbaumsaundersdistribution, author="Wolfram Research", title="{BirnbaumSaundersDistribution}", year="2016", howpublished="\url{}", note=[Accessed: 23-June-2024 ]}


@online{reference.wolfram_2024_birnbaumsaundersdistribution, organization={Wolfram Research}, title={BirnbaumSaundersDistribution}, year={2016}, url={}, note=[Accessed: 23-June-2024 ]}