# CLT and sum of gamma random variables

I am having trouble approximating the sum of gamma-distributed variables via CLR.

I know via Gamma that $$X=\sum_{i=1}^n X_i \\$$ and $$X\sim\Gamma(n\alpha,\beta) \\$$

and $$CLT: Z_{n}=\frac{\overline{X}-n\mu}{\sigma\sqrt{n}}$$

Let's say n = 300, $$X_i\sim\Gamma(0.6,0.3) \\$$ so $$X=\sum_{i=1}^n X_i \\$$, $$X\sim\Gamma(300*(0.6),0.3) \\$$

$$\mu = \frac{\alpha}{\beta} = 600.\\ \sigma^2 = \frac{\alpha}{\beta^2} = 2000\\ \sigma = 44.72\\$$

For $$Z_{n}$$, does $$n\mu$$ refer to $$300*\mu$$?

• $n\mu$ is just $600$. – Kabo Murphy Mar 27 at 7:55