# Gumbel Distribution Example

### From ReliaWiki

*This example appears in the Life Data Analysis Reference book.*

Verify using Monte Carlo simulation that if follows a Weibull distribution with and , then the follows a Gumbel distribution with and .

Let us assume that follows a Weibull distribution with and . The Monte Carlo simulation tool in Weibull++ can be used to generate a set of random numbers that follow a Weibull distribution with the specified parameters. The following picture shows the Main tab of the Monte Carlo Data Generation utility.

On the Settings tab, set the number of points to **100** and click **Generate**. This creates a new data sheet in the folio that contains random time values .

Insert a new data sheet in the folio and enter the corresponding values of the time values generated by the Monte Carlo simulation. Delete any negative values, if there are any, because Weibull++ expects the time values to be positive. After obtaining the values, analyze the data sheet using the Gumbel distribution and the MLE parameter estimation method. The estimated parameters are (your results may vary due to the random numbers generated by simulation):

Because ( ) and , then this simulation verifies that follows a Gumbel distribution with and .

Note: This example illustrates a property of the Gumbel distribution; it is not meant to be a formal proof.