[Prop] $F(x;\alpha,\beta)=F(\frac{x}{\beta};\alpha)$
Prop. Let $X$ have a gamma distribution with parameter $\alpha$ and $\beta$. Then for any $x > 0$, the cdf of $X$ is given by
\[P(X\leqslant x)=F(x;\alpha,\beta)=F(\frac{x}{\beta};\alpha)\]
where $F(\cdot;\alpha)$ is the incomplete gamma function.