Can you find the covariance matrix of a multivariate gaussian distribution if you know the means and variances of each variable?

Question

f(x,y) is a normal distribution. expval(x), expval(y), var(x) and var(y) are the only known parameter.

From this data, can the covariances be calculated?

Answer 1

No. The means have no influence on the covariances so your data give you only the diagonal elements of the covariance matrix. You also need the non-diagonal element expval$(xy)$, or equivalently the correlation coefficient between $x$ and $y$.

Let $m_x,m_y\in\mathbb R$ be given reals (the means), and $\sigma^2_x,\sigma_y^2$ be given nonnegative numbers (the variances). Let also $\rho\in[-1,1]$ be any correlation coefficient. We can construct $(x,y)$ having a Gaussian distribution with means $(m_x,m_y)$ and covariance matrix $$\begin{pmatrix}\sigma_x^2&\color{red}\rho\sigma_x\sigma_y\\\color{red}\rho\sigma_x\sigma_y&\sigma_y^2\end{pmatrix}$$ as follows: consider $u,v$ some independent, univariate standard Gaussian variables, and set $$x:=\sigma_xu+m_x$$ and $$y:=\sigma_y\left(\rho u+\sqrt{1-\rho^2}\,v\right)+m_y.$$