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While learning causal models, I came across the below assertions:

to express the conditional independence of Y and X given Z: $$ Pr(Y|X, Z) = Pr(Y|Z) $$

we can write:

$$ E(Y|X, Z) = E(Y|Z) $$ If X, Y and Z are random variables with multivariate normal distributions.

My question is shouldn't this hold for all distributions?

$$ E(Y|X, Z) = \int_{z}\int_{x} y \, dPr(y|x, z) = \int_{x} y \, d Pr(y|x) = E(Y|Z) $$

A bit rusty on the math 12 years after graduation, but what did I miss?

Any place where I can read about a proof using the multinormal property to prove the expectation equation?

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