How does one prove the formula $E[X\mid A] = \frac{E[I_A X]}{P(A)}$

Question

I'm trying to find a good way to prove the following formula on conditional expectations:

$E[X\mid A]=\frac{E[I_A X]}{P(A)}$. I am not sure how I would proceed with this. Here $I$ is an indicator of $A$ such that $I\{A\}=1$ if $A$ is true and $I\{A\}=0$ if $A$ is not true.

Answer 1

Hint:

You may be able to show

• $E[X \mid A]= E[I_A X \mid A]$
• $E[I_A X] = E[I_A X \mid A] \,P(A)$

and then rearrange, assuming $P(A)>0$