Conditional Distributions

Conditional Probability Mass Function

Definition

The conditional probability mass function is

$$f_{X|Y}(x|y)=\mathbf{P}(X=x|Y=y)=\frac{\mathbf{P}(X=x,Y=y)}{\mathbf{P}(Y=y)}=\frac{f_{X,Y}(x,y)}{f_Y(y)},$$

if $f_Y(y)>0$.

Conditional Probability Density Function

For continuous distributions, we use the same definition as the above. But one must be careful. When we compute $\mathbf{P}(X\in A|Y=y)$ in the continuous case, we are conditioning on the event $\{Y=y\}$, which as probability 0. We avoid this problem by defining things in terms of the PDF. The fact that this leads to a well-defined theory is left for more advanced courses.

Definition

For continuous random variables, the conditional probability density function is

$$f_{X|Y}(x|y)=\frac{f_{X,Y}(x,y)}{f_Y(y)}$$

assuming that $f_Y(y)>0$. Then,

$$\mathbf{P}(X\in A|Y=y)={\int }_A{f}_{X|Y}(x|y)dx.$$

From the definition of the conditional density, we see that $f_{X,Y}(x,y)=f_{X|Y}(x|y)f_Y(y)=f_{Y|X}(y|x)f_X(x)$.