Let $X_1$ and $X_2$ be independent random variables that are uniformly distributed on $\{1, \ldots, n\}$. What is the PMF of $S = X_1 + X_2$?
I'm having trouble seeing the bounds for this problem. I solved by fixing $X_2$ from $1$ to $n$ but I got the wrong answer. Can someone explain what's going on with the boundaries in this problem?
This problem generalizes the casting of a pair of dice; $S = X_1 + X_2$ takes values from $2$ to $2n$ and its PMF is determined by how many outcomes, out of a total of $n^2$, result in a given value of $S$. As an example, for two six-sided fair dice, $P(S=6) = P(S=8) = 5/36$. In general,
$$ P(S=m) = \begin{cases} \frac{m-1}{n^2} & \text{ for } 1 < m \leq n \\ \frac{2n+1-m}{n^2} & \text{ for } n < m \leq 2n \\ 0 & \text{ otherwise.} \end{cases} $$
