Let the point $(u, v)$ be chosen uniformly from the square $0\leq u\leq 1$, $0\leq v\leq 1$. Let $X$ be the random variable that assigns to the point $(u, v)$ the number $u+v$. Find the distribution function of $X$.
Now can I say that $X=u+v$ and $X$ distributed uniformly between $0$ and $2$ then should I find the cdf of the uniform $(0,2)$?
Or should I consider taking $2$ integral $1$ for $u$ and $1$ for $v$? I know the answer but I want to know the steps. :/
The answer is $F(x) = \left\{\begin{align}0,&\quad\text{when}\quad x<0\\ \tfrac{x^2}{2},&\quad\text{when}\quad0\leq x<1\\ -1+2x-\tfrac{x^2}{2}&\quad\text{when}\quad 1\leq x\leq 2\\ 1,&\quad\text{when}\quad x>2\\ \end{align}\right.$