Consider the non-linear system: $$ Z = -X + W\\ Y = X + XV. $$ Where $X$, $V$ and $W$ are mutually independent and all are $\sim U(0,1)$.
I have got some problems finding the distributions of the given exercises.
First: Determine the distribution of $Z$. $$ P(Z\leq z) = \int_{-\infty}^{\infty}\left(P_W(z+s)\right)f_X(s)ds $$ Now I know that I have to watch out with the boundaries, since $$ \begin{align} P_W(q) &= 0,\quad q < 0\\ P_W(q) &= q,\quad q \in[0,1]\\ P_W(q) &= 1,\quad q>1. \end{align} $$ So $f_Z(z) = \frac{d}{dz}\int_{-z}^{1-z}\left(P_W(z+s)\right)f_X(s)ds = 1$ if $z\in[0,1]$, with help of Leibniz. Furthermore, $F_Z(z) = -1$ if $z\in[-1,0]$ and $0$ elsewhere. Is this correct?
Second: I have to find $E[Z|Y]$. I thought of finding it the following way: $$ E[Z|Y] = E[-X + W|Y] = E[-X|Y] + E[W]\text{ since white noise and independence}\\ =E[-X|Y = y] + \frac{1}{2} = -\int_{}^{} x\frac{f_{X,Y}(x,y)}{f_Y(y)}dx + \frac{1}{2}. $$
Edit: Thank you @Did for the explanation, I too was lost in my own calculations.
Thank you for your time.