Let $(\Omega, \mathcal{F},P)$ be a probability space and $X\colon\Omega \to \mathbb{R},Y \colon \Omega \to \mathbb{R}$ be continuous random variables (i.e. random variables which have a density function. I am assuming that this implies $P(X=x)=P(Y=y)=0~\forall x,y \in \mathbb{R}$).
According to Papoulis, the conditional distribution function $F_{X|Y} = P(X \leq x | Y = y)$ is defined by considering the probability $P(X \leq x | y \leq Y \leq y + \delta y)$ and taking the limit $\delta y\to 0$. However, I do not find the derivation given there rigorous.
It is easy to write:
$P(X \leq x | y \leq Y \leq y + \delta y) = \frac{P( X \leq x, y \leq Y \leq y + \delta y)}{P(y \leq Y \leq y + \delta y)} = \frac{F_{X,Y}(x,y+\delta y) - F_{X,Y}(x,y)}{F_Y(y+\delta y) - F_Y(y)}$ from definition of $F_{X,Y}$, $F_Y$ and the fact that the point probabilities are zero.
I am not sure how to evaulate the limit:
$\lim_{\delta y \to 0} \frac{F_{X,Y}(x,y+\delta y) - F_{X,Y}(x,y)}{F_Y(y+\delta y) - F_Y(y)}.$
I have tried using the L'Hopital rule as this limit is of the form $\frac{0}{0}$ but I am not sure if that is the right direction.
Any help is much appreciated.
EDIT: Papoulis obtains a formula for the density function by differentiating the term inside the limit. i.e., $ f_{X|Y}(x,y) = \lim_{\delta y \to 0} \left(\frac{\partial F_{X,Y}(x,y+\delta y) - F_{X,Y}(x,y)}{\partial x}\frac{1}{F_Y(y+\delta y) - F_Y(y)}\right)$
I believe I should have asked for the derivation of the density function as I am afraid that the Conditional Distribution functions does not have a neat expression.
Thanks, Phanindra