Is there a name for the distribution with PDF
$$f(x; \alpha, \beta) = \alpha\beta x^{\beta - 1}e^{-\alpha x^\beta}.$$
It looks very similar to the "Type-2 Gumbel Distribution" but I'm unsure. I have derived an MLE for the parameter $\alpha$ and I want to check if it is correct!
It is otherwise known as the Weibull distribution. It is typically used in reliability analysis. The probability density function in medical statistics takes this form you have described.