Suppose $X_i$'s are iid Gaussian random variables with mean $\mu$ and variance $\sigma^2$.

The distribution of $\sum_i (X_i - \bar{X}_i)^2 / (n-1)$ isn't Chi square. What is its distribution called?

In my case, how do you determine the values of the parameters?



1 Answer 1


It's a scaled chi-square, which is generally called a gamma distribution.


The new part of the question is answered here, including an excellent hint for the derivation of the scaled chi-square result in comments.

  • $\begingroup$ Thank you. In my case, how do you determine the values of the parameters? $\endgroup$
    – Jonas
    Sep 15, 2014 at 13:15
  • $\begingroup$ Doesn't really fit in a comment. Please edit your question, and I'll respond to that in my answer. However, that now sounds rather like routine bookwork, so you should probably add the self-study tag (and read its tag wiki) $\endgroup$
    – Glen_b
    Sep 15, 2014 at 13:56
  • $\begingroup$ Yes, done. @Glen_b $\endgroup$
    – Jonas
    Sep 15, 2014 at 14:06
  • $\begingroup$ In answering it, I found the new question had been answered before, so I am now just pointing to where that part is answered. $\endgroup$
    – Glen_b
    Sep 15, 2014 at 14:33
  • $\begingroup$ In your answer in the link, $S^2 = \frac{(n-1)S^2}{\sigma^2}\cdot \frac{\sigma^2}{(n-1)}\sim \text{Gamma}(\frac{(n-1)}{2},\frac{2\sigma^2}{(n-1)})$. Should it be $S^2 = \frac{(n-1)S^2}{\sigma^2}\cdot \frac{\sigma^2}{(n-1)}\sim \text{Gamma}(\frac{(n-1)}{2},\frac{2\sigma}{\sqrt{n-1}})$ instead? $\endgroup$
    – Jonas
    Sep 16, 2014 at 19:32

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