0

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?

ECII
  • 1,791
  • 2
  • 17
  • 25
  • For more answers, please [search our site](https://stats.stackexchange.com/search?q=%22linear+combination%22+normal+score%3A2). – whuber Aug 02 '20 at 17:25
  • I did but none of the questions answers mine – ECII Aug 02 '20 at 17:50
  • Use the first link about a linear combination of random normal variables, then use induction. – Dave Aug 02 '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 02 '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 02 '20 at 18:09
  • Base case of the induction: use MGFs to show that $aY_1+bY_2$ is normal. Inductive step: show that if $Y_{n-1}$ is normal, then $cY_{n-1}+dY_n$ is normal, again by using the MGFs. – Dave Aug 02 '20 at 18:13
  • @Dave . Thanks I thought so. Still it would be nice to have an answer based on MGFs as you propose. – ECII Aug 02 '20 at 18:16
  • The answers in the duplicates are all based on the mgfs or their equivalents (cfs or cgfs). – whuber Aug 02 '20 at 19:50

0 Answers0