Gaussian variables and conditional expectation calculation

Question

Let $X,Y,Z$ be gaussian variables with zero mean. If we know the following $$E[Y\mid X=x,Z=z]=ax+bz$$ for some a and b, how can one calculate the following integral

$$\int y \int p(y \mid x,z)p(z) dzdy$$

Answer 1

By definition, $$\int yp(y \mid x,z)dy=E[Y\mid X=x,Z=z]=ax+bz$$

Hence $$\int y \int p(y \mid x,z)p(z) dzdy$$ $$=\int p(z)\int p(y|x,z).ydydz$$ $$=\int p(z) \mathbb E[Y|X=x,Z=z]dz$$ $$=\int p(z) (ax+bz)dz$$ $$=ax+b\int p(z).z dz$$ $$=ax$$

Since $z$ is Gaussian with $0$ mean.