Example 5.22

Find \(\mathbb{E}[Y]\) where \(Y\mid X \sim Gaussian(X,X^2)\) and \(X \sim Gaussian(\mu,\sigma^2)\).

First find \(\mathbb{E}_{Y \mid X}(Y \mid X)\) , which we know is simply \(X\).

Now using that, we find:

\[ \mathbb{E}[Y] = \mathbb{E}_X[\mathbb{E}_{Y|X}[Y|X]] = \mathbb{E}_X[X] = \mu \]

No need to even mess with the integrals.