My question is about how to get $\int^u_0\int^{u-y_2}_0$ and $\int^1_{u-1}\int^1_{u-y_2}$. How to find the sum of two uniform random variables by method of distribution? (without using convolution)
Let $(Y_1,Y_2)$ denote a random sample of size $n = 2$ from the uniform distribution on the interval $(0, 1)$. Find the probability density function for $U =Y_1 +Y_2$.
For 0 ≤ u ≤ 1,
$$F_U(u) = \iint _{y_1+y_2≤u}f (y_1, y_2) \, dy_1 \, dy_2 = \int^u_0\int^{u-y_2}_0(1)\,dy_1 \, dy_2 $$
For $1 < u ≤2$, (using complement)
$$F_U(u) = 1-\iint _{y_1+y_2≤u}f (y_1, y_2) \, dy_1 \, dy_2 = 1-\int^1_{u-1}\int^1_{u-y_2}(1) \, dy_1 \, dy_2 $$
I always use to sketch the graph to find the lower limit and upper limit of the integrals. But for this problem, I cannot figure out how to get $\int^u_0\int^{u-y_2}_0$ and $\int^1_{u-1}\int^1_{u-y_2}$.
Hope that someone can explain in logic and geometrically.