Let $X$ be a stochastic variable $X \sim R(0,1)$ - Thus $F_X(x) = 1$ if $x \in ]0,1[$ otherwise $0$
Let $Z$ be a stochastic variable, independent of $X$, $Z \sim b(1, 1/2)$.
$$P(Z = 0) = P(Z = 1) = 1/2$$
Let $U = X + Z$
Find distribution function of $U$. $F_U(u) = P(U \leq u) = P(X + Z \leq u)$
Prove that the distribution is $R(0,2)$
I have been trying to solve this problem for hours now, but can write the solution down. Intuitively $U$ will be distribued $R(0,2)$ since every real number $]0,1[$ $[1,2[$ has equal change by the fact $Z \sim b(1, 1/2)$
Please give me some advice and sketch out how to solve the problem.
Best regards Nicolas