Template:Characteristics of the gumbel distribution

Characteristics of the Gumbel Distribution
Some of the specific characteristics of the Gumbel distribution are the following:


 * The shape of the Gumbel distribution is skewed to the left. The Gumbel $$pdf$$  has no shape parameter. This means that the Gumbel  $$pdf$$  has only one shape, which does not change.
 * The Gumbel $$pdf$$  has location parameter  $$\mu ,$$  which is equal to the mode  $$\tilde{T},$$  but it differs from median and mean. This is because the Gumbel distribution is not symmetrical about its  $$\mu $$.
 * As $$\mu $$  decreases, the  $$pdf$$  is shifted to the left.
 * As $$\mu $$  increases, the  $$pdf$$  is shifted to the right.




 * As $$\sigma $$  increases, the  $$pdf$$  spreads out and becomes shallower.
 * As $$\sigma $$  decreases, the  $$pdf$$  becomes taller and narrower.
 * For $$T=\pm \infty ,$$   $$pdf=0.$$  For  $$T=\mu $$, the  $$pdf$$  reaches its maximum point $$\frac{1}{\sigma e}$$




 * The points of inflection of the $$pdf$$  graph are  $$T=\mu \pm \sigma \ln (\tfrac{3\pm \sqrt{5}}{2})$$  or  $$T\approx \mu \pm \sigma 0.96242$$.
 * If times follow the Weibull distribution, then the logarithm of times follow a Gumbel distribution. If $${{t}_{i}}$$  follows a Weibull distribution with  $$\beta $$  and  $$\eta $$ , then the  $$Ln({{t}_{i}})$$  follows a Gumbel distribution with  $$\mu =\ln (\eta )$$  and  $$\sigma =\tfrac{1}{\beta }$$  [32].