Normally distributed and uncorrelated does not imply independent
http://en.wikipedia.org/wiki/Normally_distributed_and_uncorrelated_does_not_imply_independent
A counterexample
The fact that two random variables X and Y both have a normal distribution does not imply that the pair (X, Y) has a joint normal distribution. A simple example is one in which Y = X if X > 1 and Y = −X if X < 1.
http://en.wikipedia.org/wiki/Multivariate_normal_distribution
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