Statistics for Maintenance Modelling: Continuous Lifetime Distributions 1

# The Gumbel (Extreme Value) Distribution¤

In many reliability situations, it is the items with shorter lifetimes that dictate policy as failures can cause system interruptions and so they need to be kept within limits. Therefore distributions tailored to modelling these poorer items are often fitted to reliability data, particularly the extreme value (or Gumbel) distribution. If a large set of items have lifetimes given by the exponential distribution, then the extreme value distribution models the smallest of these lifetimes. The probability density function for the extreme value distribution is:

Here is the scale parameter and is the location parameter. The mean of the distribution is - 0.577 .

The reliability function is:

The hazard function is:

The extreme value distribution has no shape parameter - its shape is fixed.

Click here to find out how to calculate the Gumbel distribution in Excel.