2
$\begingroup$

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?

Thanks.

$\endgroup$
2
$\begingroup$

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

http://en.wikipedia.org/wiki/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.

$\endgroup$
  • $\begingroup$ Thank you. In my case, how do you determine the values of the parameters? $\endgroup$ – Jonas Sep 15 '14 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 -Reinstate Monica Sep 15 '14 at 13:56
  • $\begingroup$ Yes, done. @Glen_b $\endgroup$ – Jonas Sep 15 '14 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 -Reinstate Monica Sep 15 '14 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 '14 at 19:32

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.