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Is it possible to have a pair of Gaussian random variables for which the joint distribution is not Gaussian?
For our upcoming exam we had to calculate the joint density of two normally distributed random variables a few times.
Say $X \sim N(0,1)$ and $Y \sim N(2,3)$.
We just assumed that $Z=(X,Y)^\top $ have a joint bivariate normal distribution $ Z \sim N\left( \left(\begin{array}{c} 0\\ 2 \end{array}\right) , \left(\begin{array}{cc} 1 & a\\ a & 3 \end{array}\right) \right) $ and calculated the only missing parameter, the covariance $a$ of the two random variables.
Now, wikipedia says that this is not always true and brings a counter example. Here is the image of the example's resulting plot.
My question: Under which conditions are they jointly normally distributed?