Differentiate the conditional CDF to get the conditional PDF Let $X$ and $Y$ be 2 continuous random variables.
Then, the conditional PDF of $X$ given $Y$ is
$f(x | y) = \frac{f(x, y)}{f(y)}$.
1) What is the definition of the conditional CDF of $X$ given $Y$?
2) How do I differentiate the conditional CDF so that I get the conditional PDF?  I think that I should differentiate with respect to $X$, but then I would not get the marginal PDF of $Y$ in the denominator of the conditional PDF.
 A: Let $X$, $Y$ be two RV's and we do not assume that they are independent. Now we are asking questions about the distribution of $X$ given $Y$.
As you stated, the conditional PDF of $X$ given $Y$ is
$$f\left(X=x|Y=y\right)=\frac{f\left(X=x,Y=y\right)}{f\left(Y=y\right)}$$
For the ease of understanding, we can define a new continuous variable $Z_{y}$ that is equal in distribution to $X$ for any given $Y=y$, that is:
$$P\left(Z_{y}<z\right)=P\left(X<z|Y=y\right)\,\forall\,z,y$$
and thus we get:
$$f\left(Z_{y}=z\right)=\frac{\partial}{\partial z}P\left(Z_{y}<z\right)=\frac{\partial}{\partial z}P\left(X<z|Y=y\right)=f\left(X=z|Y=y\right)\,\forall\,z,y$$
Note that since we are conditioning on $Y$, when observing $Z_{y}$ we can relate to $y$ as a constant. Now we can answer your questions quite easily using our previous knowledge in univariate probability:
$$P\left(X<x|Y=y\right)=P\left(Z_{y}<x\right)=\int_{-\infty}^{x}f\left(Z_{y}=u\right)du=\int_{-\infty}^{x}f\left(X=u|Y=y\right)du=\frac{1}{f\left(Y=y\right)}\cdot\int_{-\infty}^{x}f\left(X=u,Y=y\right)du$$
And the other way around:
$$f\left(X=x|Y=y\right)=f\left(Z_{y}=x\right)=\frac{\partial}{\partial x}P\left(Z_{y}<x\right)\\=\frac{1}{f\left(Y=y\right)}\cdot\frac{\partial}{\partial x}\int_{-\infty}^{x}f\left(X=u,Y=y\right)du=\frac{f\left(X=x,Y=y\right)}{f\left(Y=y\right)}$$
The major point I'm trying to make here is that for an RV $X$ and for every given $Y=y$ we can define a new RV $Z_{y}\overset{d}{=}\left\{ X|Y=y\right\}$  which we can relate to as a univariate RV with $y$ as a parameter. Everything else will be the same as we learned in basic probability.
