Let X and Y be independent identically distributed random variables with a uniform distribution [0,1]. Denoted by Z = X + Y.
(a) Find the density of f(Z).
(b) Show that for every x belonging to [0,1], the events Z > 1 and (-x < Z <= 1 + x) are independent.
I calculated the density of f(Z) as:
0 <= Z <= 1: z
1 < Z <= 2: 2 - z
0: otherwise