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Jan 13, 2016 at 0:08 comment added Douglas Zare I was mistaken about interpolation. Your statement seems reasonable to me now.
Jan 12, 2016 at 20:58 comment added math-Student Douglas, you are right, the equality can not imply Gaussianity. However, I am looking for a necessary and sufficient condition on $P_{XY}$ with Gaussian $Y$ such that we have the equality. Can you also elaborate a bit on "then interpolate to get equality".
Jan 12, 2016 at 3:20 history asked math-Student CC BY-SA 3.0