Given $X_1...X_n\stackrel{iid}{\sim} N(\mu,\sigma^2)$ and $U=\sum_{i=1}^n (X_i-\overline{X})^2$, why is $U\sim\sigma^2 \chi_{n-1}^2$ ?

And what would be the distribution of $V=\sum_{i=1}^n (X_i-\mu)^2$ ?

  • $\begingroup$ $X_i$ are independent right? And also, is this a self-study question? If so, please add a tag. $\endgroup$
    – psarka
    Jan 3, 2019 at 11:57
  • $\begingroup$ yeah, sorry. fixed it $\endgroup$ Jan 3, 2019 at 12:35
  • $\begingroup$ Hint: $X_1-\mu, X_2-\mu, \ldots, X_n-\mu$ are iid $N(0,\sigma^2)$ random variables. What does your book have to say about this distribution? Nothing? How about considering $(X_1-\mu)/\sigma, (X_2-\mu)/\sigma, \ldots, (X_n-\mu)/\sigma$ which are iid $N(0,1)$ random variables? $\endgroup$ Jan 3, 2019 at 14:43
  • $\begingroup$ See stats.stackexchange.com/questions/385032/… and the links posted there. $\endgroup$ Jan 3, 2019 at 15:00


Browse other questions tagged or ask your own question.