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I am looking for the lower bound of $$P(XY>u)$$ where $X$,$Y$ are correlated centred Gaussian random variables.

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Zero. $ $ $ $ $ $ –  Did Feb 14 '13 at 14:30
You can find some helpful results from the book - \emph{Probability Distributions Involving Gaussian Random Variables: A handbook for engineers, scientist and mathematician}. –  Bo LI Sep 18 '13 at 19:29

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