What does E(X + Y) mean?

Question

I am working on some problems in probability theory and keep getting stuck on some of the concepts regrading expected values. I understand that if you have one dice roll you would have a distribution of $X$ with values taking ie. $P(X=1) = {1 \over 6}$, but how do you interpret this when you are considering two die rolls? Understanding that these are independent you would essentially get the same distribution.

Question: What is the distribution of $(X + Y)$ and how would you calculate $E(X + Y)$ if $X$ represents the first roll and $Y$ represents the second roll?

Is this just the same as $E(X) + E(Y)$?

Part 2: If you are trying to find $E(2X - 2)$ do you subtract two from each different value? For example $[2 * 1 * {1 \over 6} - 2] + [2 * 2 * {1 \over 6} -2]$...

Answer 1

No matter whether X and Y are dependent or independent, $$E(X+Y)=E(X)+E(Y)$$

Coming to $E(2X-2)$, it can be easily written as $E(2X) - E(2)$ which in turn equals to $2E(X)-2$.

Answer 2

The expected value operator is "linear"

$E[X+Y] = E[X] + E[Y]\\ E[aX] = aE[X]\\ E[2X + Y + 2] = 2E[X] + E[Y] + 2$