Given $N\geq2$, we choose two different numbers randomly from $\{0,1,...,N-1\}$, and assign them to the random variables X and Y respectively. How can we calculate a PGF for the sum $X+Y$?
I noticed X and Y are dependent, so $g_{X+Y}(s)\neq g_{X}(s)g_{Y}(s)$. However, we know $Y|X$ is distributing uniformly on $\{0,1,...,N-1\}$ \ $\{X\}$.
Now, I started by $\mathbb{E}[s^{X+Y}]=\mathbb{E}[s^{X}s^{Y}]=\mathbb{E}[s^{X}\mathbb{E}[s^{Y}|X]]$. But I don't know how to continue.
I found that $\mathbb{E}[Y|X]=\frac{N^2-2X}{2(N-1)}$, but it doesn't seem to be helpful for calculating the PGF of X+Y.
Remarks:
- $\mathbb{E}[var] = $ Expected Value Of...
- $g_v(s)=$ The PGF of the random variable V.