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

Corollary 4.3.1

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?

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