$X\in Unif[0,4]$ and $Y\in Unif[0,1]$. $Z=X+Y$ Find the distribution of $Z$ and $Fz(4.5)$

Question

I used $$\int_{0}^{4}\int_{0.5}^{4.5-x} \frac{1}{4} \,dy \,dx$$ and the answer was $2$. Is this correct? How do I to find the official distribution?

Any help will be appreciated.

Answer 1

The following figure depicts the red areas denoted by $A_u$ (for the different values of a $u$) that have to be multiplied by $\frac14$, the constant value of the density function.

$$F_{X+Y}(u)=P(X+Y<u)=\frac14\iint_{A_u}\ 1\ dxdy=$$ $$=\frac14\begin{cases}0&\ \text{ if }& \ u\le 0\\ \frac{u^2}2 &\ \text{ if }& \ 0<u\le 1\\ u-\frac12&\ \text{ if }& \ 1<u\le 4\\ 4-\frac12(5-u)^2&\ \text{ if }& \ 4<u\le 5\\ 4&\ \text{ if }& \ u>5.\\ \end{cases}$$

Substituting $4.5$ we get $\frac{31}{32}$.