I believe I'm having trouble interpreting the definition of a conditional density function. The definition from Wikipedia states the following:
The conditional probability density function of $Y$ given the occurrence of the value $x$ of $X$ is given by $$ f_{Y}(y|X=x) = \frac{f_{X,Y}(x,y)}{f_Y(y)} $$
It is my understanding that this is the same as $$ f_{Y|X}(y|x) = \frac{f_{X,Y}(x,y)}{f_Y(y)}, $$ another common form with a slight notational difference.
With this in mind, is the conditional density function a function of two variables? Everywhere I look seems to define the conditional density only as a function of $y$. I understand that $Y$ is being conditioned on a given value of $X$, but the value that $X$ takes on is still represented by a variable $x$. This would lead me to believe that both variables ($x$ and $y$) must be accounted for by the function, implying $f_{Y|X}(y|x): \mathbb{R}^2 \mapsto [0,\infty)$, not $f_{Y|X}(y|x): \mathbb{R} \mapsto [0,\infty)$.
Perhaps I'm misunderstanding the scope of the definition. If someone could point me in the right direction here I would greatly appreciate it.
The value of $X=x$ is known, and can be interpreted as a "parameter" (in a functional sense). It gives different parametrizations of the functions, and indeed when one talks of $f_{Y\mid X}(y \mid x)$ one generally refers to a family of conditional distributions (one for every value in the support of $X$). However, one must not forget that every member of this family is parametrized by a known value of $X$. Therefore, for density purposes, $f_{Y\mid X}(y \mid x)$ is a function only of $Y$.