1.
Exploratory Data Analysis
1.3. EDA Techniques 1.3.6. Probability Distributions 1.3.6.6. Gallery of Distributions


Probability Density Function 
The BirnbaumSaunders distribution is also commonly known as the
fatigue life distribution. There are several alternative
formulations of the BirnbaumSaunders distribution in the literature.
The general formula for the probability density function of the BirnbaumSaunders distribution is \( f(x) = \left (\frac{\sqrt{\frac{x\mu} {\beta}} + \sqrt{\frac{\beta} {x\mu}}} {2\gamma (x\mu)} \right) \phi \left( \frac{\sqrt{\frac{x\mu} {\beta}}  \sqrt{\frac{\beta} {x\mu}}} {\gamma} \right) \hspace{.2in} x > \mu; \gamma, \beta > 0 \) where γ is the shape parameter, μ is the location parameter, β is the scale parameter, \(\phi\) is the probability density function of the standard normal distribution, and \(\Phi\) is the cumulative distribution function of the standard normal distribution. The case where μ = 0 and β = 1 is called the standard BirnbaumSaunders distribution. The equation for the standard BirnbaumSaunders distribution reduces to \( f(x) = \left (\frac{\sqrt{x} + \sqrt{\frac{1} {x}}} {2\gamma x} \right) \phi \left (\frac{\sqrt{x}  \sqrt{\frac{1} {x}}} {\gamma} \right) \hspace{.2in} x > 0; \gamma > 0 \) Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function. The following is the plot of the BirnbaumSaunders probability density function.


Cumulative Distribution Function 
The formula for the cumulative distribution
function of the BirnbaumSaunders distribution is
\( F(x) = \Phi(\frac{\sqrt{x}  \sqrt{\frac{1} {x}}} {\gamma}) \hspace{.2in} x > 0; \gamma > 0 \) where \(\Phi\) is the cumulative distribution function of the standard normal distribution. The following is the plot of the BirnbaumSaunders cumulative distribution function with the same values of γ as the pdf plots above.


Percent Point Function 
The formula for the percent point
function of the BirnbaumSaunders distribution is
\( G(p) = \frac{1} {4} \left[\gamma \Phi^{1}(p) + \sqrt{4 + (\gamma \Phi^{1}(p))^{2}}\right]^{2} \) where \(\Phi^{1}\) is the percent point function of the standard normal distribution. The following is the plot of the BirnbaumSaunders percent point function with the same values of γ as the pdf plots above.


Hazard Function 
The BirnbaumSaunders hazard function can
be computed from the BirnbaumSaunders probability density and cumulative
distribution functions.
The following is the plot of the BirnbaumSaunders hazard function with the same values of γ as the pdf plots above.


Cumulative Hazard Function 
The BirnbaumSaunders cumulative hazard
function can be computed from the BirnbaumSaunders cumulative
distribution function.
The following is the plot of the BirnbaumSaunders cumulative hazard function with the same values of γ as the pdf plots above.


Survival Function 
The BirnbaumSaunders survival
function can be computed from the BirnbaumSaunders cumulative
distribution function.
The following is the plot of the BirnbaumSaunders survival function with the same values of γ as the pdf plots above.


Inverse Survival Function 
The BirnbaumSaunders inverse survival
function can be computed from the BirnbaumSaunders percent point
function.
The following is the plot of the gamma inverse survival function with the same values of γ as the pdf plots above.


Common Statistics 
The formulas below are with the location parameter equal to zero and
the scale parameter equal to one.


Parameter Estimation  Maximum likelihood estimation for the BirnbaumSaunders distribution is discussed in the Reliability chapter.  
Comments  The BirnbaumSaunders distribution is used extensively in reliability applications to model failure times.  
Software 
Some general purpose statistical software programs, including
Dataplot, support at least
some of the probability functions for the BirnbaumSaunders distribution.
Support for this distribution is likely to be available for
statistical programs that emphasize reliability applications.
The "bs" package implements support for the BirnbaumSaunders distribution for the R package. See
