Given random variables X and Y such that X ~ N(x , x 2 ) and Y|X ~ N(0 + 1x, 2 ), Show that the joint distribution of (X, Y) is bivariate normal.

LaTeX for the distributions:

$$ \begin{array}{c} x \sim \mathrm{N}\left(\mu_{x}, \sigma_{x}^{2} ight) \\ y \mid x \sim \mathrm{N}\left(\beta_{0}+\beta_{1} x, \sigma^{2} ight) \end{array} $$

Please be as detailed as possible as I'm not too familiar with this topic.