Given the location and scale parameters of a Gumbel distribution for variable X, how does one calculate the mean and variance of X^2? I am working with predictive models for wind speeds, which have been given as Gumbel distributions. I need to convert the wind speeds to wind pressures using the formula:
$Pressure = Density * Velocity^2$
So my question is how do I determine the distribution parameters for the wind pressure given the distribution of the velocity? 
Also, can I assume that $X^2$ also has a Gumbel distribution? The problem is fairly easy if I can just assume that it has a normal distribution and that:
$E(X^2) = \mu^2 + \sigma^2$
$var(X^2) = E(X^4)-(E(X^2))^2$
 A: The Gumbel cumulative distribution function (CDF) with location parameter $\alpha$ and scale parameter $\beta$ is
$$F_{\alpha, \beta}(x) = 1 - \exp\left(-\exp\left(\frac{x-\alpha}{\beta}\right)\right).$$
When $X$ has this distribution, the CDF of $X^2$ by definition equals
$$F_{X^2}(t) = \Pr(X^2 \le t) = \Pr(-\sqrt{t}\le X \le \sqrt{t}) = F_{\alpha, \beta}(\sqrt{t}) -  F_{\alpha, \beta}(-\sqrt{t}) \\
=\exp\left(-\exp\left(\frac{-\sqrt{t}-\alpha}{\beta}\right)\right) - \exp\left(-\exp\left(\frac{\sqrt{t}-\alpha}{\beta}\right)\right)$$
for $t\ge 0$ (and is equal to $0$ otherwise).  This is not a Gumbel distribution (it cannot possibly be, because Gumbels always assign positive probability to negative values), but at least it explicitly provides a formula in terms of the parameters $\alpha$ and $\beta$.
For working with the density function (PDF), differentiate this CDF with respect to $t$.
A: If you just want the mean and variance of $Y = c X^2$ where $X \sim {\rm Gumbel}(\alpha,\beta)$, then observe that ${\rm E}[Y] = c{\rm E}[X^2] = c({\rm Var}[X] + {\rm E}[X]^2)$, so all we need to know here is the mean and variance of the Gumbel distribution.  For higher moments, we would need $${\rm Var}[Y] = {\rm E}[Y^2] - {\rm E}[Y]^2 = c^2 {\rm E}[X^4] - c^2({\rm Var}[X] + {\rm E}[X]^2)^2,$$ so we require the fourth raw moment of $X$.  This can be obtained via the MGF; e.g., $${\rm E}[X^4] = \frac{d^4}{dt^4}\left[M_X(t)\right]_{t=0}.$$  Refer to the Wikipedia entry for the Gumbel distribution.
