# 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:

$E(X^2) = \mu^2 + \sigma^2$

$var(X^2) = E(X^4)-(E(X^2))^2$

• You can't simply assume that $X^2$ is Gumbel, nor can you necessarily assume either $X$ or $X^2$ is normal (though $X^2$ might not be too far off. But the formulas you give at the end should apply; they're not based on normality. Mathworld gives the kurtosis for the Gumbel so you should be able to get the variance of the square from that. – Glen_b May 21 '14 at 14:59
• As it turns out, $X^2$ is often highly non-normal, and can be either left or right skew, but is sometimes vaguely normalish -- if the scale is not too large relative to the location parameter (e.g. coefficient of variation around 0.125-0.15 or or thereabouts). But since you can deal with the distribution reasonably easily, I wouldn't bother with what's at best a pretty rough approximation even then. – Glen_b May 21 '14 at 23:05

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 $t$.

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.