# 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?

• $\bar X = \frac1n \sum X_i$ so you can use the moment generating function again Aug 5, 2019 at 10:38
• Ok I thought I would be easier:( Aug 5, 2019 at 10:46

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.