What is the joint distribution of sample mean and sample variance of normal distribution? $X_i \sim N( \mu,\sigma^2)$, define $\overline X =\dfrac{1}{n} \sum\limits_{i = 1}^n X_i $, $S^2 = \dfrac{1}{n - 1}\sum\limits_{n = 1}^n \left( {X_i - \overline X} \right)^2$. What is the distribution of 
$$
\sqrt n \left( \begin{array}{c}
\overline X  - \mu \\
S^2 - \sigma ^2
\end{array} \right)
$$

The distribution of the mean and variance of a normal rv is very well known:
$$\sqrt n \left( \begin{array}{c}
\overline X  - \mu \\
{S^2} - {\sigma ^2}
\end{array} \right) \sim \ \left(\begin{array}{c}
\mathcal{N}(0,1) \\
\sigma^2\left(\frac{\sqrt{n}\chi^2_{n-1}}{n-1}-1\right)
\end{array} \right)$$
