I've been reading different textbooks and I'm so confused by the notation of conditional probability density functions(p.d.f's).
Please correct my understandings below.
Setting: First of all, we use $\omega$ to denote the elements in the sample space of one experiment, use $\mu$ to denote the elements in the sample space of anther experiment. We assign random variables X and Y respectively to these experiments.
Understanding:
The subscription of p.d.f refers to the corresponding random variable.
Say, $f_x$ is the p.d.f of random variable X, namely the derivative of $F(X(\omega)\leq x)$ on all values of x. Following the pattern, I would say $f_{x|y}(x|y)$ is still a p.d.f on X, but the difference is that it's calculated by taking derivative of $F(X(\omega)\leq x| Y(\mu) = y)$ on all values of x, namely cumulative distribution of X assuming $Y(\mu)$ equals a fixed value of y.
Only by thinking this way, I can justify why $f_{x|y}(x|y)\equiv f(x|y) \equiv f_{x|y}(x)$. If my way of thinking is wrong, please interpret the different notation in the previous sentence and state why are they the same.
Also, $f_x(x)$ means exactly the same as $f_X(x)$, which is derivative of $F(X(\omega)\leq x)$.