# Parametric Binomial Example - Demonstrate MTTF

This example appears in the Life Data Analysis Reference book.

In this example, we will use the parametric binomial method to design a test that will demonstrate ${\displaystyle MTTF=75\,\!}$ hours with a 95% confidence if no failure occur during the test ${\displaystyle f=0\,\!}$. We will assume a Weibull distribution with a shape parameter ${\displaystyle \beta =1.5\,\!}$. We want to determine the number of units to test for ${\displaystyle {{t}_{TEST}}=60\,\!}$ hours to demonstrate this goal.

The first step in this case involves determining the value of the scale parameter ${\displaystyle \eta \,\!}$ from the ${\displaystyle MTTF\,\!}$ equation. The equation for the ${\displaystyle MTTF\,\!}$ for the Weibull distribution is:

${\displaystyle MTTF=\eta \cdot \Gamma (1+{\frac {1}{\beta }})\,\!}$

where ${\displaystyle \Gamma (x)\,\!}$ is the gamma function of ${\displaystyle x\,\!}$. This can be rearranged in terms of ${\displaystyle \eta \,\!}$:

${\displaystyle \eta ={\frac {MTTF}{\Gamma (1+{\tfrac {1}{\beta }})}}\,\!}$

Since ${\displaystyle MTTF\,\!}$ and ${\displaystyle \beta \,\!}$ have been specified, it is a relatively simple matter to calculate ${\displaystyle \eta =83.1\,\!}$. From this point on, the procedure is the same as the reliability demonstration example. Next, the value of ${\displaystyle {{R}_{TEST}}\,\!}$ is calculated as:

${\displaystyle {{R}_{TEST}}={{e}^{-{{({{t}_{TEST}}/\eta )}^{\beta }}}}={{e}^{-{{(60/83.1)}^{1.5}}}}=0.541=54.1\%\,\!}$

The last step is to substitute the appropriate values into the cumulative binomial equation. The values of ${\displaystyle CL\,\!}$, ${\displaystyle {{t}_{TEST}}\,\!}$, ${\displaystyle \beta \,\!}$, ${\displaystyle f\,\!}$ and ${\displaystyle \eta \,\!}$ have already been calculated or specified, so it merely remains to solve the binomial equation for ${\displaystyle n\,\!}$. The value is calculated as ${\displaystyle n=4.8811,\,\!}$ or ${\displaystyle n=5\,\!}$ units, since the fractional value must be rounded up to the next integer value. This example solved in Weibull++ is shown next.

The procedure for determining the required test time proceeds in the same manner, determining ${\displaystyle \eta \,\!}$ from the ${\displaystyle MTTF\,\!}$ equation, and following the previously described methodology to determine ${\displaystyle {{t}_{TEST}}\,\!}$ from the binomial equation with Weibull distribution.