Difference between conditional expectation E(Y|X) and E(Y|X=x)

Question

I'm trying to solve this problem and have a few questions.

So for part i), I don't know how to find this conditional expectation without pure intuition that it seems to be $\frac{3+x}{2}$

part ii) I'm conceptually confused by this because the right-hand side is just X instead of "X=x" (so just a general X versus a specific number). How would I calculate this?

part iii) I'd need to know how to do part i and ii to describe this

part iv) I could just use the law of iterated expectations here, where E(Y) = E[E(Y|X)]. Pretty straightforward.

Any help would be great!

Answer 1

$$ \operatorname E(Y\mid X=x) = \frac{x+3} 2. $$ Since $x$ is not a random variable, neither is $\dfrac{x+3} 2.$

But $\operatorname E(Y\mid X) = \dfrac{X+3} 2$ is a random variable, since $X$ is a random variable.