# Probability - expected value

The random variable $$X$$ takes on values -2, 0 and 2 with probabilities 1/4, 1/2 and 1/4 respectively. Find $$\text{E}(X)$$ and $$\text{Var}(X)$$.

Till this part, it was easy enough.

Then the question continues, the random variable $$Y$$ is defined by $$Y = X_1 + X_2$$, where $$X_1$$ and $$X_2$$ are two independent observations of $$X$$. Find $$\text{Var}(Y)$$ and $$\text{E}(Y + 3)$$.

What I did: All possible combinations of $$X_1$$ and $$X_2$$ also turns out to be $$Y \in \{-2,0,2\}$$. But I don't know what the probabilities will be? Will it be the same?

Here's a hint: you could do this the long way, by figuring out every value of $$Y$$ and its probability and then computing the mean and variance from that. But you know that $$Y$$ is the sum of two independent random variables (we would usually say that $$X_1$$ and $$X_2$$ are independent copies of $$X$$, not observations), and you know the means and variances of those. There's a quicker way to the answer using those facts.