Conditional Probability Mass Function

For Discrete Random Variable $X$, $Y$, $P((X,Y)\in A|(X,Y)\in B)={\sum }_{(x,y)\in A}P(X=x,Y=y|(X,Y)\in B)$ Where:

• $P(X=x,Y=y|(X,Y)\in B)=\frac{P(\{X=x,Y=y\}\cap \{(X,Y)\in B\})}{P((X,Y)\in B)}$ represents the conditional PMF of $X,Y$ given $(X,Y)\in B$

One-Variable PMF

$p_{X|Y}(x|y)=P(X=x|Y=y)=\frac{P(X=x,Y=y)}{P(Y=y)}=\frac{p_{X,Y}(x,y)}{p_Y(y)}$

Defining Joint PMF

$P(X=x,Y=y)=P(X=x)P(Y=y|X=x),\forall x,y$ $p_{X,Y}(x,y)=p_X(x)p_{Y|X}(y|x),\forall x,y$