Location and Scale parameter of a Gumbel 2 type distribution

Question

I have a random variable X with Gumbel 2 type distribution. Here the density:

$f_X(x)$=$\beta$$e^{\beta(x-log(\alpha))-e^{\beta(x-log(\alpha))}}$

I have to show that the Gumbel is a location-scale distribution and after that have to find $\mu$ and $\sigma$... Unfortunately I can't find so many references on the web, the few cases always refer to the Gumbel 1 type. Anyone can help me?

Answer 1

From the probability density it appears that the support is from $-\infty$ to $\infty$. Using Mathematica one finds the following:

d = ProbabilityDistribution[beta Exp[beta (x - Log[alpha]) - 
      Exp[beta (x - Log[alpha])]],
  {x, -Infinity, Infinity}];
mean = Expectation[x, x ~ d, Assumptions -> {beta < 0, alpha > 0}]
variance = FullSimplify[Expectation[x^2, x ~ d, 
    Assumptions -> {beta < 0, alpha > 0}] - mean^2]

The mean is $\frac{\gamma }{\beta }-\log (\alpha )$ and the variance is $\frac{4 \gamma \log (\alpha )}{\beta }-2 \log ^2(\alpha )-\frac{12 \gamma ^2+\pi ^2}{6 \beta ^2}$ where $\gamma$ is Euler's constant ($\simeq 0.577216$).