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.