# Example Using a Distribution to Approximate the CDF

*This example appears in the System Analysis Reference book*.

Consider the system shown next.

Components [math]A\,\![/math] through [math]E\,\![/math] are Weibull distributed with [math]\beta =1.2\,\![/math] and [math]\eta =1230\,\![/math] hours. The starting and ending blocks cannot fail. Compute a single Weibull distribution approximation for the system.

**Solution**

Assume that the system can be approximated by a 2-parameter Weibull distribution with [math]\beta =2.02109\,\![/math] and [math]\eta =1123.51\,\![/math]. In BlockSim, this is accomplished by representing the entire system as one distribution. To do this, click the **Distribution fit** area on the diagram's control panel, as shown next.

This opens the Distribution Estimator window, which allows you to select a distribution to represent the data.

When you analyze the diagram, BlockSim will generate a number of system failure times based on the system's reliability function. The system's reliability function can be used to solve for a time value associated with an unreliability value. For example, consider an unreliability value of [math]F(t)=0.11\,\![/math]. Using the system's reliability equation, the corresponding time-to-failure for a 0.11 unreliability is 389.786 hours. The distribution of the generated time values can then be fitted to a probability distribution function.

When enough points have been generated, the selected distribution will be fitted to the generated data set and the distribution's parameters will be calculated. In addition, if ReliaSoft's Weibull++ is installed, the fit of the distribution can be analyzed using a Weibull++ standard folio, as shown in the next figure. It is recommended that the analyst examine the fit to ascertain the applicability of the approximation.