Partition Theorem for expectations - why only dependent on value of X and not Y?

Question

Hi there,

In my textbook above, there is a theorem stating that the expectation of $E [Y | X]$ is equal to $E[Y]$.

However, the textbook also states that this theorem only depends on the value of $X$. Why not the value of $Y$?

Thanks very much

Answer 1

In my textbook above, there is a theorem stating that the expectation of E[Y|X] is equal to E[Y].

That is often known as the Law of Total Expectation, or the Tower Property:

$\qquad\mathsf E(Y)=\mathsf E(\mathsf E(Y\mid X))$.

However, the textbook also states that this theorem only depends on the value of X. Why not the value of Y?

Your textbook should state that the conditional expectation only depends on the value of $X$.

This is for the same reason the unconditional expectation is not dependent on $Y$.

  $\mathsf E(Y)$ is a constant.   It does not vary with $Y$ because it is the expected value (or mean value) of that random variable.

  Similarly, $\mathsf E(Y\mid X)$ a function of $X$.   It is the expected value of $Y$ as a function of a given value for $X$.