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I read in a textbook that when two gaussian variables are uncorrelated, then they are statistically independent? How can I prove that?

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    You can't prove that because it is not true in general. Uncorrelated _jointly Gaussian_ random variables are independent. If the random variables are Gaussian but not jointly Gaussian, then they could be uncorrelated and yet be dependent. There are standard examples. Search this web site for other answers to this problem.2012-01-20

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