Conditional Distributions

We describe one variable given another.

Conditional pdf of $Y$ given $X$

$$f_{Y\mid X}(y\mid x)=\frac{fX,Y(x,y)}{fX(x)}$$

The conditional pdf is the joint pdf over the marginal pdf

The variable we are conditional on is treated as a constant. In this case it is $x$.

The conditional pdf must integrate to 1

$$\int f_{Y\mid X}(y\mid x)dy=1$$