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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. – Dilip Sarwate Jan 20 '12 at 22:24

If $X$ and $Y$ are jointly gaussian, and uncorrelated, you can show that $$f_{XY}(x,y) = f_X(x)f_Y(y);$$ this assures independence.

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But be sure to check out this recent question on stats.SE where a related issue regarding independence is being discussed. – Dilip Sarwate Jan 21 '12 at 1:09

Here is a counterexample.

Here is the definition of "'jointly' normally distributed". This article states, but I'm not sure it proves, that jointly normally distributed random variables are independent if they are uncorrelated.

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