Question: Suppose that you have the random variables $X$ and $Y$ with known distribution functions. How would you derive the distribution function of $Z$, where $Z$ is defined as some combination of $X$ and $Y$.
For instance, suppose that $Z = \dfrac{X}{Y+1}$, would we have $F_Z = \dfrac{F_X}{F_Y+1}$? I can't find the theory that should explain this.
When $Z=\dfrac{X}{1+Y}$ then
$$\begin{align}\mathsf P(Z\leqslant z) &= \mathsf P(\tfrac{X}{1+Y}\leqslant z) \\ &=\mathsf P(X\leqslant z(1+Y)\cap Y>-1)~+~\mathsf P(X\geqslant z(1+Y)\cap Y<-1)\end{align}$$
Next would be an application of the Law of Total Probability, the form of which depends on what is the joint distribution for $X,Y$, and in particular, whether $Y$ is a continuous or discrete random variable.