Expected Value of the product of an indicator R.V. and continuous R.V.

Question

I have an indicator random variable $X \in \{0,1\}$ and a continuous random variable $Y$. I am looking for $E[XY]$. Intuitively, it seems $E[XY] = P(X=1)E[Y|X=1]$, but am having difficulty identifying the law that shows this. Any thoughts?

Answer 1

It's the Law of Iterated Expectation; nothing more.

$\begin{align}\mathsf E(XY) ~=~ & \mathsf E(\mathsf E(XY\mid X)) \\[1ex] ~=~& \mathsf P(X=1)\,\mathsf E(XY\mid X=1) + \mathsf P(X=0)\,\mathsf E(XY\mid X=0) \\[1ex] ~=~& \mathsf P(X=1)\,\mathsf E(Y\mid X=1) + \mathsf P(X=0)\,\mathsf E(0\mid X=0) \\[1ex] ~=~ & \mathsf P(X=1)\,\mathsf E(Y\mid X=1)\\[2ex]\Box\qquad\qquad\quad &\end{align}$