This is part of a short series on the common life data distributions.

The Birnbaum-Saunders distribution is a univariate continuous distribution. This short article focuses on 7 formulas of the Birnbaum-Saunders Distribution. This distribution was designed to model the Miner’s rule, thus allowing for non-constant fatigue cycles through accumulated damage.

If you want to know more about fitting a set of data to a distribution, well that is in another article.

It has the essential formulas that you may find useful when answering specific questions. Knowing a distribution’s set of parameters does provide, along with the right formulas, a quick means to answer a wide range of reliability related questions.

## Parameters

The scale parameter, β, is equal to the median.

The shape parameter is designated as α. As α gets smaller the distribution becomes more symmetrical around the scale parameter, β.

## Probability Density Function (PDF)

When t > 0 then the probability density function formula is:

$$ \displaystyle\large \begin{array}{l}f\left( t \right)=\frac{\sqrt{{}^{t}\!\!\diagup\!\!{}_{\beta }\;}+\sqrt{{}^{\beta }\!\!\diagup\!\!{}_{t}\;}}{2\alpha t}\\f\left( t \right)=\frac{\sqrt{{}^{t}\!\!\diagup\!\!{}_{\beta }\;}+\sqrt{{}^{\beta }\!\!\diagup\!\!{}_{t}\;}}{2\alpha t}\phi \left( t \right)\\\text{where }\phi \left( t \right)\text{ is the standard normal pdf and:}\\z=\frac{\sqrt{{}^{t}\!\!\diagup\!\!{}_{\beta }\;}-\sqrt{{}^{\beta }\!\!\diagup\!\!{}_{t}\;}}{\alpha }\end{array}$$

A plot of the PDF provides a histogram-like view of the time-to-failure data.

## Cumulative Density Function (CDF)

F(t) is the cumulative probability of failure from time zero till time t. Very handy when estimating the proportion of units that will fail over a warranty period, for example.

$$ \displaystyle\large \begin{array}{l}

F\left( t \right)=\Phi \left( \frac{\sqrt{{}^{t}\!\!\diagup\!\!{}_{\beta }\;\ }-\sqrt{{}^{\beta }\!\!\diagup\!\!{}_{t}\;\ }}{\alpha } \right) \\

F\left( t \right)=\Phi \left( z \right) \\

\end{array}$$

## Reliability Function

R(t) is the chance of survival from from time zero till time t. Instead of looking for the proportion that will fail the reliability function determine the proportion that are expected to survive.

$$ \displaystyle\large \begin{array}{{l}}

R\left( t \right)=\Phi \left( \frac{\sqrt{{}^{\beta }\!\!\diagup\!\!{}_{t}\;\ }-\sqrt{{}^{t}\!\!\diagup\!\!{}_{\beta }\;\ }}{\alpha } \right) \\

R\left( t \right)=\Phi \left( -z \right) \\

\end{array}$$

## Conditional Survivor Function

The m(x) function provides a means to estimate the chance of survival for a duration beyond some known time, t, over which the item(s) have already survived. What the probability of surviving time x given the item has already survived over time t?

$$ \displaystyle\large \begin{array}{{l}}

m\left( x \right)=R\left( x|t \right)=\frac{\Phi \left( -{z}’ \right)}{\Phi \left( -z \right)} \\

\text{where} \\

{z}’=\frac{\sqrt{{}^{\left( t+x \right)}\!\!\diagup\!\!{}_{\beta }\;\ }-\sqrt{{}^{\beta }\!\!\diagup\!\!{}_{\left( t+x \right)}\;\ }}{\alpha } \\

\end{array}$$

## Mean Residual Life

This is the cumulative expected life over time x given survival till time t.

$latex u\left( t \right)=\frac{\int_{t}^{\infty }{\Phi \left( -z \right)dx}}{\Phi \left( -z \right)}$$

## Hazard Rate

This is the instantaneous probability of failure per unit time.

$$ \displaystyle\large h\left( t \right)=\frac{\sqrt{{}^{t}\!\!\diagup\!\!{}_{\beta }\;\ }-\sqrt{{}^{\beta }\!\!\diagup\!\!{}_{t}\;\ }}{2\alpha t}\left[ \frac{\phi \left( z \right)}{\Phi \left( -z \right)} \right]$$

## Cumulative Hazard Rate

This is the cumulative failure rate from time zero till time t, or the area under the curve described by the hazard rate, h(x).

$$ \displaystyle\large H\left( t \right)=-\ln \left[ \Phi \left( -z \right) \right]$$

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