Suppose $U \sim Unif(0,1)$ and $Z \sim Unif(U,3+U)$. How can I find the pdf for $U + Z$?

Question

Suppose $U \sim Unif(0,1)$ and $Z\mid U \sim Unif(U,3+U)$. I would like to find the pdf for $U + Z$, which in my process on $Z$ is a continuation of $U$. Is there a straightforward way to derive this?

Answer 1

Looks like we have another uniform random variate, call it $V$ such that:

$$V \sim \text{Unif}(0,3)$$ and $$Z \sim U \star V \quad \text{(convolution)}$$ so that $$Z= U+V.$$

So now, $$U+Z = 2U + Z.$$

Let $$W=2U$$ so $$W \sim \text{Unif}(0,2).$$

$$U+Z = W+V \sim \text{Unif}(0,2) \star \text{Unif}(0,3).$$

The distribution will be a trapezoid:

$$ \left\{ \begin{array}{ll} \frac{y}{6}, & 0 < y \le 2 \\ \frac{1}{3}, & 2<y\le 3 \\ \frac{5-y}{6}, & 3<y<5 \end{array} \right.$$