Conditional Probability Density Function

Let $X$ and $Y$ be two-dimensional random variables defined simultaneously on a sample space $S$. The joint probability function of $X$ and $Y$ is $ P(X=x, Y=y)$. Again the marginal probability function of $X$ and $Y$ are $g(x)$ and $h(y)$, where $g(x)>0$ and $h(y)>0$. Then the conditional probability function of $X$ given $Y=y$ is given by,

\[g(x/Y=y)=\frac{P(X=x,Y=y)}{h(y)},h(y)>0.\]

Similarly, the conditional probability function of $Y$  given $X=x$ is given by

\[h(y/X=x)=\frac{P(X=x,Y=y)}{g(x)},g(x)>0.\]

If $X$ and $Y$ are two dimensional continuous random variable, then

\[g(x/Y=y)=\frac{f(x,y)}{h(y)},h(y)>0.\]

and \[h(y/X=x)=\frac{f(x,y)}{g(x)},g(y)>0.\]