Let $X$ and $Y$ have a uniform distribution on the set of points with integer coordinates in $S = \{(x,y):0\leqq x\leqq7, x \leqq y \leqq x+2\}$. That is, $f(x,y) = \frac{1}{24}\in S$, and both x and y are integers.
Find $f_X(x)$
I can compute in my head that $f_X(x)=\frac{1}{8}$ (which is the correct answer). However I would like to solve this using the definition: $$f_X(x)=\sum_y f(x,y) = P(X=x), x \in S$$
I am not quite sure how to write the summation in this format given that for each $x$ there are three possible values for $y$.