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Let's define two random variables $x_1\sim\mathcal{N}(\mu_1,\sigma^2_1)$, $x_2\sim \mathcal{N}(\mu_2,\sigma^2_2)$. Also, $x=\binom{x_1}{x_2}$. We know $$\Sigma_{xx}=\left( \begin{array}{cc} \sigma_1^2 & \rho\sigma_1\sigma_2 \\ \rho\sigma_1\sigma_2 & \sigma_2 ^2\\ \end{array} \right),$$ where $\rho$ is the correlation between $x_1$ and $x_2$.

What happen with $\Sigma_{xx}$ when $x_1$ and $x_2$ have other distribution? (For example Uniform, Exponential, Gamma, ChiSquared, etc.)

Is there any reference using other type of distributions to obtain $\Sigma_{xx}$?

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The covariance matrix does not have such a nice expression in general:

https://en.wikipedia.org/wiki/Covariance_matrix#Definition

Note also that there are no unique bivariate versions of the distributions you mentioned (exponential, chi-squared ...).

You might also be interested in looking at Copulas:

https://en.wikipedia.org/wiki/Copula_(probability_theory)

The bivariate Gaussian copula contains a parameter that quantifies the dependence between the marginals.

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