covariance matrix in bivariate distribution

Question

I struggle to understand how exactly you get the covariance matrix in a bivariate normal distribution. The reason is probably that I have no idea how to obtain it at all. In the exercise I have I already have $\sigma_{11}^2$ and $\sigma_{22}^2$ as well as $\rho_{12}$. I also have the two $\mu$ values. But with the formulas I find around the internet I'm not able to figure out how to calculate the covariance matrix for that distribution. How do I start here? Thanks in advance.

Answer 1

You can read more here. You will get:

$$ \Sigma = \left( \begin{array}{cc} \sigma_{1}^2 & \rho_{12} \sigma_{1} \sigma_{2} \\ \rho_{12} \sigma_{1} \sigma_{2} & \sigma_{2}^2 \end{array} \right) $$