Template:GumbelDistribution

The Gumbel Distribution

The Gumbel distribution is also referred to as the Smallest Extreme Value (SEV) distribution or the Smallest Extreme Value (Type 1) distribution. The Gumbel distribution is appropriate for modeling strength, which is sometimes skewed to the left (e.g., few weak units fail under low stress, while the rest fail at higher stresses). The Gumbel distribution could also be appropriate for modeling the life of products that experience very quick wear out after reaching a certain age.

The pdf of the Gumbel distribution is given by:

[math]\displaystyle{ \begin{align} f(t)= & \frac{1}{\sigma }{{e}^{z-{e^z}}} \\ z= &\frac{t-\mu }{\sigma } \\ f(t)\ge & 0,\sigma \gt 0 \end{align} }[/math]

where,

[math]\displaystyle{ \begin{align} \mu = & \text{location parameter} \\ \sigma = & \text{scale parameter} \end{align} }[/math]

The Gumbel distribution and its characteristics are presented in detail in the chapter The Gumbel/SEV Distribution.