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If I know that $A \sim \mathcal N(0,\sigma^2_A)$ and $B \sim \mathcal N(0,\sigma^2_B)$ and that $A$ and $B$ have a correlation coefficient of $\rho$, does this mean that it must be that case that

$\begin{bmatrix} A \\ B \end{bmatrix} \sim \mathcal N\left( \begin{bmatrix} 0 \\ 0 \end{bmatrix}, \begin{bmatrix} \sigma^2_A & \rho \sigma_A \sigma_B \\ \rho \sigma_A \sigma_B & \sigma_B^2 \end{bmatrix} \right) \>?$

Or is that just one possibility?

And what about for other distributions?

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It's just one possibility. Wikipedia suggests a counter example, in which two dependent normal random variables are not bivariate normal.

If $A$ and $B$ are independent, then the vector $(A,B)$ is bivariate normal.

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