Method of moment generating function with Gamma Distribution

Assume that $\displaystyle Y_1, \ Y_2, ... \ Y_n$ is a sample space of size n from a gamma distribution population with $\displaystyle \alpha = 2$ and $\displaystyle \beta$ unknown.

Us the method of moment generating function to show that $\displaystyle 2 \sum_{i=1}^n \frac{Y_i}{\beta}$ is a pivotal quantity and has a $\displaystyle \chi^2$ distribution with 4n df.

I know that the moment generating function of a Gamma Distribution is $\displaystyle (1-\beta t)^{-\alpha}$.

I attempted to replicate http://www.mathhelpforum.com/math-he...-function.html, but got nowhere. I figure it has to like:

$\displaystyle (1-2t)^{-4n}$ since that's the mgf of a $\displaystyle \chi^2$ distribution with 4n df.