Assume that ${(X, Y)}$ has joint density ${ c e^{-(1+ x^{2})(1+ y^{2} ) } }$, where ${ c }$ is properly given. How can we prove that ${X}$ and ${Y}$ are Gaussian random variables, but that ${(X, Y)}$ is not a Gaussian vector.
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Just check that the joint density does not have the form of a bivariate normal density. The marginals can be found by integration. – StubbornAtom Mar 25 '20 at 05:10
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There seems to be something seriously wrong. The marginals are nor normal. Did you perhaps say joint density instead of joint characteristic function? – Kavi Rama Murthy Mar 25 '20 at 05:18
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Yes, I mean the joint density. Could you please show how to calculate the exponential integral? I tried to do that but got stuck. – FrankZ Mar 25 '20 at 05:37
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The marginals are not normal, the conditional distributions are. Recheck the exercise. – StubbornAtom Mar 25 '20 at 06:09
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https://stats.stackexchange.com/q/184414/119261 – StubbornAtom Mar 25 '20 at 06:15
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Very helpful stuff. Thanks! – FrankZ Mar 25 '20 at 06:30