Conditional Expectance

Question

I have the following formula to prove but I don't understand anything from notation. I know conditional expectance but what is Y1(z=i)] Do you understand it ? And if you do so, can you please explain to me what this equality tells in simple english. If I understand the problem to prove then I can start working on it thanks.

Answer 1

$Y\mathbf1_{\{Z=i\}}$ stands for the product of random variable $Y$ and random variable $\mathbf1_{\{Z=i\}}$.

The second one takes value $1$ if $\omega\in\{Z=i\}$ (or equivalently if $Z(\omega)=i$) and takes value $0$ otherwise.

So $Y\mathbf1_{\{Z=i\}}$ is prescribed by $\omega\mapsto Y(\omega)$ if $Z(\omega)=i$ and $\omega\mapsto0$ otherwise.

This calculation of expectation of $Y$ must be made "under condition $Z=i$".

So we are simply not interested values taken by $Y$ for $\omega\notin\{Z=i\}$.