Definition (Conditional pmf)

Given two discrete RVs $X$ and $Y$ with joint pmf $p(x,y),x\in\mathscr{X},y\in\mathscr{Y}$, the conditional pmf of $x$ given that $Y=y$ is denoted by $p_{X|Y}(x|y)$ and defined as $$p_{X|Y}(x|y)=\frac{P(X=x,Y=y)}{P(Y=y)}=\frac{p_{xy}(x,y)}{p_y(y)}, \ x\in\mathscr{X}$$

Definition (Conditional Probability Density Function)

Given two continuous RVs $X$ and $Y$ with joint pdf $f_{xy}(x,y),x\in\mathscr{X},y\in\mathscr{Y}$, the conditional pdf of $X$ given that $Y=y$ is denoted by $f_{X|Y}(x|y)$ and defined as $$f_{X|Y}(x|y)=\frac{f_{xy}(x,y)}{f_y(y)}, \ x\in\mathscr{X}, \ f(y)>0$$