# Proof that the sample mean is normally distributed [duplicate]

I am studying the book of Larsen and Marx and stumbled upon

I can prove that $$\bar{Y}=\frac{1}{n} \sum_{i=1}^{n} Y_{i}$$ and $$Var(\bar{Y})=\frac{\sigma ^{2}}{n}$$ but how would I go to show that $$\bar{Y}$$ is normally distributed? Why normal? Should I use the Normal MGF?

– whuber
Aug 2 '20 at 17:25
• I did but none of the questions answers mine
– ECII
Aug 2 '20 at 17:50
• Use the first link about a linear combination of random normal variables, then use induction.
– Dave
Aug 2 '20 at 17:53
• They all answer this question, because $\bar Y$ is a linear combination of the $Y_i$ with coefficients $1/n.$
– whuber
Aug 2 '20 at 17:55
• @whuber I understand this but why should the distrubution of $\bar{Y}$ be normal? How do you prove normality of the resulting distribution?
– ECII
Aug 2 '20 at 18:09