Sum of normally distributed random variables / moment generating functions1 I consider $n$ independent random variables with $$X_1 \sim N(\mu_1, \sigma_1^2),\ldots,X_n \sim N(\mu_n, \sigma_n^2).$$ 
I was able to show that $$ \sum_{i=1}^n a_i X_i +b \sim N(\sum_{i=1}^n a_i \mu_i+b, \sum_{i=1}^n a^2_i \sigma_n^2 ).$$
I managed to do that using moment-generating functions $X_1+X_2 \sim N(\mu_1+\mu_2, \sigma_1^2 +\sigma_2^2) $ and $M_{X_1+X_2}(t) = exp(\mu_1+\mu_2)t + \frac{(\sigma_1^2 +\sigma_2^2) \cdot t}{2}) $
How can I show now that $\bar{X}\sim N( \mu, \frac{\sigma^2}{n})$
Can somebody give me a hint?

You can just use the previous result twice:
First derive the distribution of $Y= \sum_i X_i$ and then the one of $Z = \frac{1}{n}Y$, both are particular cases of the first result.
