Let $X, Y, Z$ be independent random variables each with the uniform distribution on the interval $[0,1]$.

(a) Show that $X+Y$ has density function

$$f_{X+Y}(u)= \begin{cases}u & \text { if } 0 \leq u \leq 1 \\ 2-u & \text { if } 1 \leq u \leq 2 \\ 0 & \text { otherwise }\end{cases}$$

(b) Show that $P(Z>X+Y)=\frac{1}{6}$.

(c) You are provided with three rods of respective lengths $X, Y, Z$. Show that the probability that these rods may be used to form the sides of a triangle is $\frac{1}{2}$.

(d) Find the density function $f_{X+Y+Z}(s)$ of $X+Y+Z$ for $0 \leqslant s \leqslant 1$. Let $W$ be uniformly distributed on $[0,1]$, and independent of $X, Y, Z$. Show that the probability that rods of lengths $W, X, Y, Z$ may be used to form the sides of a quadrilateral is $\frac{5}{6}$.