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Given a vector $\mu = \{\mu_x \mu_y \mu_z\} $ and an equivalent covariance matrix $\Sigma$, I used the following in Mathematica to compute the conditional expectation $\mathbf{E}(Y|Y+Z)$.

Expectation[ 
 y \[Conditioned] y + z == y0 + z0 , {x, y, z} \[Distributed] 
  MultinormalDistribution[mu, sigma]]

Which produced the correct result.

As I try something a bit more tricky like $\mathbf{E}(Y|(Y+Z)^2)$ with the following:

Expectation[ 
 y \[Conditioned] (y + z)^2 == (y0 + z0)^2 , {x, y, z} \[Distributed] 
  MultinormalDistribution[mu, sigma]]

Mathematica does not output a result.

I imagine Mathematica is capable of this computation so I must be inputting something in incorrectly. Is this possible to compute in Mathematica knowing Mean and Covariance Matrix?

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    – bbgodfrey
    Sep 30 '21 at 1:00

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