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I am working on a problem and hit a wall. I don't need the whole problem answered. just this part.

$X_1,...X_n \sim N(\theta, \theta^2)$, what is the distribution of ${\sum_1^nX_i^2}/n$?

It's messing me up that these aren't centered. If they weren't scaled...I could divide by $\theta^2$ and it would be some scaled chi-squared. I can't figure out the noncenterd though. could anyone help?

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How about you first center and scale them.

$$ \dfrac{X_i - \theta}{\theta} \sim N(0,1).$$

Now, $$ \dfrac{1}{n}\sum_{i}\dfrac{X_i^2}{\theta} - 1 \sim \chi^2_{(n)}.$$

Now you can try and move constants around. Hint: $\chi^2_{(n)}$ can be written as another distribution.

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If $Y\sim N_n (\mu,\Sigma)$ then $Y^\prime \Sigma^{-1} Y \sim \chi^2_n (\mu^\prime \Sigma^{-1} \mu)$. You just have to rearrange this to get the right form.

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