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I read in the answers to this question (Bivariate Normal Conditional Variance) that

$$\Bbb E(Y\mid{X=x})=\mu_Y + \rho \frac{\sigma_Y}{\sigma_X}(x-\mu_X)$$

By symmetry I would say that

$$\Bbb E(X\mid{Y=y})=\mu_X + \rho \frac{\sigma_X}{\sigma_Y}(y-\mu_Y)$$

Is this correct ?

Next question, referring to the picture below, does taking expectations along the red lines (conditionning on X) or along the blue lines (conditioning on Y) yield the same line in this picture ?

enter image description here

I can see that these equations are different, but that is because they are in another basis, do these equations represent the same line ?

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Is this correct ?

Yes. Symmetry is applicable.

There is not much else to say.

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  • $\begingroup$ Are both lines the principal axis of the ellipses? $\endgroup$ – user0009 Dec 31 '17 at 12:35
  • $\begingroup$ see the picture that I added $\endgroup$ – user0009 Dec 31 '17 at 13:06

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