# Normal Distribution with unknown mean and unknown variance [duplicate]

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: $$$$\begin{split} \frac{X_i - \mu}{\sigma} & \sim N(0, 1) \\ \sum_{i = 1}^{n} \left ( \frac{X_i - \mu}{\sigma} \right )^2 & \sim \chi_{n}^2 \\ \sum_{i = 1}^{n} \left ( \frac{X_i - \bar{X}_n}{\sigma} \right )^2 & \sim \chi_{n - 1}^2 \\ \frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} & \sim N(0, 1) \\ \frac{\bar{X}_n - \mu}{\sigma' / \sqrt{n}} & \sim t_{n - 1} \end{split}$$$$

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?