So, I'm not sure how to solve this problem:
$X \in \Gamma(a,b)$. Show that $$\frac{X-E[X]}{\sqrt{Var(X)}} \rightarrow^{d} N(0,1)$$ as $a \rightarrow \infty$
Use the central limit theorem.
I've come to $\frac{X-ab}{b\sqrt{a}}$, but my feeling is that this is not the same as CLT
$$S_{n} = X_{1} + X_{2} + ... + X_{n}, n \geq 1$$ $$\frac{S_{n}-n\mu}{\sigma \sqrt{n}} \rightarrow^{d} N(0,1)$$ as $n \rightarrow \infty$
since $S_{n}$ is a sum of r.v.'s and $X$ is a single r.v..