Converting a Conditional Expectation into an Unconditional one

Question

Consider two random variables $V_1$ and $V_2$, where $V_1$ is continuous and $V_2$ is binary.

In a paper, I found that the following equality holds: \begin{align*} \mathbb{E}\left[V_1|V_2=1\right]=\mathbb{E}\left[\frac{V_2}{\mathbb{E}[V_2]}V_1\right] \end{align*} I remember that I have proved that this equality holds without any assumption about two years ago.

But, I cannot come up with the proof again.

So, does that equality holds always? or other assumptions are needed?

Answer 1

In general, for any random variable $X$ and an event $A$ with $P(A) > 0$, $E[X|A]$ is defined as (see this question, this question and this question for related discussions): \begin{align} E[X|A] = \frac{E[XI_A]}{P(A)}. \tag{1}\label{1} \end{align}

Now taking $X = V_1$ and $A = [V_2 = 1]$ in $\eqref{1}$, and using the fact that $V_2$ is binary (so that $I_{[V_2 = 1]} = V_2$ and $P(V_2 = 1) = E[V_2]$) completes the proof.