Let $\overline X$ be a binormal random variable with distribution $N_{\overline X}(\overline m, \Sigma)$ where,
$\overline X = \left( \begin{array}{c} x \\ y \end{array} \right)$, $\overline m = \left( \begin{array}{c} m_1 \\ m_2 \end{array} \right)$$ \Sigma = \left( \begin{array}{cc} \sigma_1^2 & \rho\sigma_1\sigma_2 \\ \rho\sigma_1\sigma_2 & \sigma_2^2 \end{array} \right)$
Show that the marginal distribution is also normal with: $p(x_1)=N_{x_1}(m_1,\sigma_1^2)$ and that the conditional distribution is normal with: $p(x_1|x_2)=N_{x_1}(m_1+\frac{\rho\sigma_1(x_1-m_2)}{\sigma_2};\sigma_1^2(1-\rho^2))$.
Here is the begining of my calculation:
I am not sure how to continue from here. Shoud I use the series expansion of $e^x$ in $$\int e^{-\frac{y^2}{\sigma_y^2}}dy$$
Am I in the right direction?