Let $\pmatrix{X\\Y\\Z} \sim \mathcal N\left(\pmatrix{0\\0\\0},\pmatrix{1&\rho_{12}&\rho_{13}\\\rho_{12} & 1 & \rho_{23}\\\rho_{13}&\rho_{23}&1}\right)$.
Compute $E[XY|Z]$.
I have no idea how to do it. Any ideas?
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Have you tried using the definition of conditional expectation? Integrate with respect to the conditional distribution. The latter depends solely on the joint distribution, which is known... – mbe May 31 '17 at 18:32
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$(X,Y)$ given $Z$ has a bivariate normal distribution. – StubbornAtom Oct 24 '21 at 13:28