The Notation of Conditional Expectation

Question

Wikipedia told me the fomula for $E(X|Y=y)$, where $y$ is an particular number. But in my homework, the question is to find $E(X|Y)$, and $Y$ has two values, $1$ and $-1$, under different conditions. Is it equal to $E(X|Y=1)+E(X|Y=-1)$?

Answer 1

Is it equal to $E(X\mid Y=1) + E(X\mid Y=-1)$?

No, $E(X\mid Y) = E(X\mid Y=y)$ where $y$ is unspecified. It is a function of $y$ and is a random variable. By definition,

$$E(X\mid Y) = \sum_{x}{xP(X=x\mid Y=y)}.$$

In your particular case, since $Y$ only takes values $1$ and $-1$, one way to specify $E(X\mid Y)$ is to evaluate $E(X\mid Y=1)$ and $E(X\mid Y=-1)$, which are numbers: say $a = E(X\mid Y=1)$ and $b = E(X\mid Y=-1)$:

$$E(X\mid Y) = \begin{cases} a, & \text{if $Y=1$} \\ b, & \text{if $Y=-1$} \\ \end{cases}$$