I got this question as a challenge question from my lecturer and I have no idea how to even approach it. I need to show that $$ f(x) = \frac{\alpha}{\beta} \left( \frac{x}{\beta} \right)^{\alpha - 1} e^{- (x / \beta)^{\alpha} }, \hspace{0.5cm} x \geq 0 $$ is a probability density function.
The lecturer gave a hint and said that I can use a change of variable to evaluate the integral. I also know that to prove a function is a PDF that when integrated from $0$ to $\infty$, it should equal $1$. however, I don't even know where to being or what change of variable I am meant to do.
The original integral can be written as follows
$$\int_{0}^{+\infty}\frac{\alpha x^{\alpha-1}}{\beta^{\alpha}}\cdot e^{-(\frac{x}{\beta})^\alpha}dx=\int_{0}^{+\infty} e^{-(\frac{x}{\beta})^\alpha}d[(\frac{x}{\beta})^\alpha]=1$$