Uniform Distribution and Distribution function technique Let $X_1$ and $X_2$ be independent random variables having the uniform density with $\alpha = 0$ and $\beta = 1$. Find expressions for the function $Y =X_1 + X_2$.
(a)$y \le 0$
(b)$0<y<1$
(c)$1<y<2$
(d)$y\ge2$
I'm thinking $f(x_1)=f(x_2) = 1$ for $0 \le x_1\le 1$ and $0 \le x_2\le 1$. I suppose this implies that $f(x_1,x_2) = 1$ so $F(x_1,x_2) = \int_0^y\int_0^{y-x_2} f(x_1,x_2) dx_1dx_2 $ Then I would need $\frac{d}{dy} F(x_1,x_2)$ to find an appropriate $f(y)$
I already have the answers to these in my book but I don't know how to come upon them as they don't correspond at all to what I have here. Where is my fault?

Or, alternatively, without words...

