Question on use of indicator function

Question

For my Financial Mathematics course I have the following exercise (with solutions):

I don't really understand the start of the solution of (B). More specifically I do not understand why the $1_{\{X=\alpha\}}$ is added, isn't that redundant because of the nature of the Sigma Sum?

Answer 1

$\mathbb E[Y \mid X =\alpha]$ is a number. Conditioning on $X=\alpha$ is just like conditioning on any other event; then we take the expected value.

On the other hand, $\mathbb E[Y \mid X]$ is a random variable. Specifically, it is a random variable that:

• Equals $\mathbb E[Y \mid X=-1]$ when $X=-1$;
• Equals $\mathbb E[Y \mid X=1]$ when $X=1$.

The indicator functions are also random variables, and they have the exact effect we wanted. When $X=1$, for example, only the $1_{X=1}$ function is nonzero, and it contributes a $\mathbb E[Y\mid X=1]$ term for the sum.

Without the indicator functions $1_{X=\alpha}$ in the sum, we'd just get the number $\mathbb E[Y \mid X=-1] + \mathbb E[Y\mid X=1]$, not a random variable that depends on the value of $X$.