General Gamma distribution clarification

Question

In my notes defining a general gamma distribution where the "waiting time" is not necessarily an integer it states the following for $\alpha >0$, $\beta >0$: $$\int_0^{\infty}\exp(-\beta t)t^{\alpha -1 }dt=\frac {\Gamma (\alpha)} {\beta ^ \alpha}.$$ I know that the Gamma function is defined as $$\Gamma (\alpha) = \int_0^{\infty}y^{\alpha-1}e^{-y}dy,$$ but I am unsure how these two statements are connected.

Answer 1

Simply make the next change in your integral $t = s/\beta$ and you will have it!

For clarity

$$\int_{0}^{\infty}\textrm{exp}(-\beta t)t^{\alpha-1}dt = \int_{0}^{\infty}\textrm{exp}(-s)\left(\frac{s}{\beta}\right)^{\alpha-1}d\left(\frac{s}{\beta}\right)=\frac{\Gamma(\alpha)}{\beta^{\alpha}}$$