Random variable for sum of probability densities. Given two Random Variables: $X$ with probability density $f_X(x)$ and $Y$ with probability density $f_Y(y)$, I want to understand the random variable $Z$ such that $Z$ has probability density $f_X(z) + f_Y(z)$. Here, I'm allowing for negative probability densities, and $f_X$ and $f_Y$ do not necessarily integrate to 1. 
NB: I'm not looking for the formula for the pdf of $Z = X+Y$, which was answered many times in other posts.
I want to know if there is a way to express $Z$ in terms of $X$ and $Y$. For example, can I find constants $c_{ij}$ to say $Z = \sum_{ij} c_{ij} X^iY^j$?
Thank You!
EDIT: After reading some comments, it seems like posing this question in terms of $X$ and $Y$ as random variables doesn't really make sense if $f_X(x)$ and $f_Y(y)$ are not always positive and do not integrate to 1. So I want to ask the simpler version of this question. If $f_X(x)$ and $f_Y(y)$ are proper probability density functions, and we have some random variable $Z$ with pdf: $pf_X(z) + (1-p)f_Y(z)$ where $p \in [0,1]$, is there a way to express $Z$ in terms of $X$ and $Y$ (like $Z = \sum_{ij} c_{ij} X^iY^j$)?
($X$ and $Y$ are independent) 
 A: The edited question is amenable to standard probability theory: $Z$ is a mixture of $X$ and $Y$. More precisely, let $B$ be Bernoulli with parameter $0<p<1$, independent of $X$ and $Y$. If $X$ and $Y$ have PDFs $f_X$ and $f_Y$ respectively, then $Z=BX+(1-B)Y$ has PDF $f(z)=pf_X(z)+(1-p)f_Y(z)$. 
(Though it may not be what you desired to have the coefficients $c_{ij}$ themselves random...)
A: Note that the distribution or density functions of a function of your $\ X\ $ and $\ Y\ $ will, in general, depend not merely on their individual density functions, $\ f_X\ $ and $\ f_Y\ $, but on their joint distribution or density function, $\ f_{XY}\ $, assuming that this latter exists.  With that qualification, if $\ \varphi:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}\ $ is any function at all such that $\ \varphi(X,Y)\ $ has a density $\ f_\varphi\ $, then the random variable
$$
\int_{-\infty}^{\varphi(X,Y)}f_\varphi(t)dt
$$
will be uniformly distributed.
I have to confess that I have no idea what it means for a random variable $\ Z\ $ to have a density function $\ f_X+f_Y\ $ which integrates to a value of $2$, rather than $1$, but if you define
$$
F_\text{zinv}(y)=\inf \left\{x\left| \int_{-\infty}^x\left(f_X(t)+f_Y(t)\right) dt\ge 2y\right.\right\}\ ,
$$
then the random variable
$$
Z=F_\text{zinv}\left(\int_{-\infty}^{\varphi(X,Y)}f_\varphi(t)dt\right)
$$
will have density function $\ \frac{f_X+f_Y}{2}\ $.
