I am interested to know how one can derive, from first principles, the Weibull distribution. As I understand it, the Weibull distribution $$D_{w}(x) = \frac{k}{\lambda} \left(\frac{x}{\lambda}\right)^{k-1} e^{{(-x/\lambda)}^{k}}$$ is a generalised form of the Rayleigh distribution $$ D_{R}(x) = \frac{x}{\sigma^{2}} e^{-x^{2}/(2\sigma^{2})} $$ when $k = 2$ and $\lambda = \sqrt{2}\sigma$.
The Rayleigh distribution can be derived from first principles very easily. Consider a vector, $\mathbf{v} = (x, y)$ where $x$ and $y$ are Gaussian distributed. If the overall magnitude of $\mathbf{v}$ is considered a Rayleigh distribution emerges.
Is there a similar argument one can make to derive the Weibull distribution? OIf not, how can one derive the Weibull from first principles?
Motivation: I know one can just introduce a shape parameter. I am trying to et an understanding of how Weibull's emerge in nature with some physical intuition. Very much in the same way as one can think how the Rayleigh emerges.
