Let $X$ denote the outcome of a standard six-sided dice. Let $Y$ denote the outcome of a standard six-sided dice, but we only roll $Y$ provided that $X \geq 3$. If $X \leq 2$, then set $Y = 0$. Compute $E(X + Y)$ and $\text{Var}(X + Y)$.
So I computed $E(X)$ as follows: $E(X) = \frac{2}{6}(1.5) + \frac{4}{6}(4.5) = 1.5$.
Also $E(Y) = \frac{2}{6}(0) + \frac{4}{6}(3.5) = 2.333$.
Then I summed to get $3.83$ for $E(X + Y)$ but this is wrong.
Can someone help explain why?
Your $E(X)$ is calculated incorrectly. $X$ is just a single die roll, which should have expected value $3.5$. Period. It doesn't matter that we change the rules for the next roll based on the value of $X$ - $X$ itself is still just a single fair die roll.
Now, looking closer at that - you have a correct formula for $E(X)$. You just didn't evaluate it properly.
Your $E(Y)$ is correct, and the expected values do sum.