In this Probability Density Function, am I doing it right?

Question

Consider a model consisting of random variables $X,Y$ and $Z: Y=Xw+Z$, where $Z∼U[−1,1]$.

Answer 1

Your question is right, other than talking about their independence ( which they are not having the equality $Y=\omega X + Z$ means they cannot be ! ).

For writing it down you can do it this way (although your intuition is fine, it may not be for more complicated problems) :

Let $F_{Y|X=x}$ be the conditional CDF, and $f_{Y|X=x}$ the conditional PDF.

Recall $\frac{d}{da}F_{Y|X=x}(a)=f_{Y|X=x}(a)$ (fundamental theorem of calculus + definitions of CDF and PDF). Now calculate

For $a \in [\omega x-\frac{1}{2},\omega x+\frac{1}{2}]$

\begin{align} F_{Y|X=x}(a) =& \mathbb{P}(Z \leq a-\omega x) \\ =& a-\omega x - \frac{1}{2} \end{align}

hence $\frac{d}{da}F_{Y|X=x}(a)=1=f_{Y|X=x}(a)$ for $a \in [\omega x-\frac{1}{2},\omega x+\frac{1}{2}]$.