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# BirnbaumSaundersDistribution

 BirnbaumSaundersDistribution represents the Birnbaum-Saunders distribution with shape parameter and scale parameter .
• The cumulative distribution function for value in a Birnbaum-Saunders distribution is given by , where is the CDF for the standard normal distribution.
Probability density function:
Cumulative distribution function:
Mean and variance:
Median:
Probability density function:
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Cumulative distribution function:
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Mean and variance:
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Median:
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 Scope   (7)
Generate a set of pseudorandom numbers that are Birnbaum-Saunders distributed:
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 :
The limiting value:
Kurtosis depends only on the shape parameter :
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:
 Applications   (3)
The lifetime in hours of a component has a Birnbaum-Saunders distribution with and . 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 a component A follows a Birnbaum-Saunders distribution with and , while the failure rate of component B is . 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 Birnbaum-Saunders distribution tends to fail early:
The lifetime of a device has a Birnbaum-Saunders 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 :
Parameter influence on the CDF for each :
Birnbaum-Saunders distribution is closed under scaling by a positive factor:
If has a Birnbaum-Saunders distribution, then also has a Birnbaum-Saunders distribution:
Birnbaum-Saunders distribution is related to NormalDistribution:
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