I am new to probabilities and I am trying to solve the following problem:
We randomly select two numbers from $0$ to $N-1$ for $N = 1, 2, \dots$. What is the probability function of $P(n)$, where $n$ is the difference of the two numbers selected, given that $n$ greater or equal to zero? Prove that the probability is normalized to one.
Intuitively, I found that $P(n)$ is $\frac{N-n}{N^2}$. Is it possible to reach this result by applying a theorem?
Also how can be possible, the requested probability to sum to $14$, given that the we select only the positive deferences? I am thinking for example, that if I roll two dice, then the differences I can have vary from $-5$ to $5$, and the sum of their probabilities is one. How can be possible the sum of a subtotal of the possible events (as the positive differences) to sum to one? Do I miss something?
Thanks a lot!