Conditional expectation of two independent variables

Question

If E(X) = E(X|Y), does this mean that the expected value of variable X is equal to the sum of expected values of X given all values of Y?

Answer 1

The equation does not occur in this form in the file you linked to. The equation in the file is

$$ E(X\mid Y)=E(X) $$

for $X$, $Y$ independent. While equations are formally symmetric, they also function as sentences that make a statement about the left-hand side, namely that it is equal to the right-hand side; the left-hand side is, as it were, the subject of the sentence corresponding to an equation. The statement here is that the left-hand side, which is a random variable, is in fact constant and takes the constant value on the right-hand side.

To answer your question: No, this does not mean that $E(X)$ is the sum of $E(X\mid Y)$ for all $Y$. It means that $E(X\mid Y)$ is a constant random variable with the constant value $E(X)$.

Something similar to what you may have had in mind with the sum is that $E(X)$ is the expected value of $E(X\mid Y)$, with the expectation taken over all values of $Y$. This, however, doesn't require $X$ and $Y$ to be independent. The law of total expectation states that

$$ E(E(X\mid Y))=E(X) $$

and doesn't require independence of $X$ and $Y$. Of course it also holds in the present case, with $X$ and $Y$ independent:

$$ E(E(X\mid Y))=E(E(X))=E(X)\;. $$

To make this all more concrete, you could think e.g. of two fair six-sided dice, and let $X$ and $Y$ be the numbers they show. Then we have $E(X)=\frac72$ and also $E(X\mid Y)=\frac72$, as knowing which side the second die shows tells us nothing about which side the first die shows: $X$ and $Y$ are independent.