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I'm new to Statistics, so bear with me. Let $X_1, \ldots. X_n$ be iid $N(\mu, \sigma^2)$ random variables, where $\mu$ and $\sigma^2$ are unknown. Let $\bar{X}_n = \sum_{i = 1}^n X_i / n$ and $\sigma' = \sqrt{\sum_{i = 1}^n (X_i - \bar{X}_n)^2 / (n - 1)}$. We know the following: $$ \begin{equation} \begin{split} \frac{X_i - \mu}{\sigma} & \sim N(0, 1) \\ \sum_{i = 1}^{n} \left ( \frac{X_i - \mu}{\sigma} \right )^2 & \sim \chi_{n}^2 \\ \sum_{i = 1}^{n} \left ( \frac{X_i - \bar{X}_n}{\sigma} \right )^2 & \sim \chi_{n - 1}^2 \\ \frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} & \sim N(0, 1) \\ \frac{\bar{X}_n - \mu}{\sigma' / \sqrt{n}} & \sim t_{n - 1} \end{split} \end{equation} $$

What seems to be missing from the list is the distribution of

$$ \sum_{i = 1}^{n} \left ( \frac{X_i - \bar{X}_n}{\sigma'} \right )^2. $$ Is this not important or not useful?

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marked as duplicate by whuber distributions Jan 25 at 13:38

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