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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?

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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.

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