Understand conditional expectation of indicator functions.

Question

I am new to conditional probability/expectation and am trying to get my head around the following equation.

$$\int \operatorname{var}\left(\mathbb{E}\left(1_{\{Y \leq t\}} \mid X\right)\right) d \mu(t),$$ where $\mu(t)$ is the distribution of Y, Y=aX+bZ, and X and Z are standard normal I.i.d. .

I had a difficult time thinking about $\mathbb{E}\left(1_{\{Y \leq t\}} \mid X\right)$. Would it be equal to $1_{\{ax+bE(Z) \leq t\}}$ or $P(aX+bE(Z)\leq t)$? How should I think of it?

Any help is greatly appreciated.

Answer 1

$X,Z$ are independent and identically standard normal, and $Y=aX+bZ$

If $a,b$ are positive real constants then, the conditional expectation is a function of the random variable $X$, and that is:

$$\begin{align}\mathsf E(\mathbf 1_{Y\leq t}\mid X)&=\mathsf E(\mathbf 1_{Z\leq (t-aX)/b}\mid X)\\&=\Phi((t-aX)/b)\end{align}$$

Where $\Phi$ is the standard-normal cumulative distribution function.