Definition (Marginal probability mass function)

Let $X$ and $Y$ be two random variables. The marginal distributions can be computed from their joint pmf $p(x,y)$, as a result the marginal pmf of $X$ is $$p_X(x)=\sum_{y\in\mathscr{Y}}p(x,y), \ x\in\mathscr{X}$$ and the marginal pmf of $Y$ is $$p_Y(y)=\sum_{x\in\mathscr{X}}p(x,y), \ y\in\mathscr{Y}$$

Definition (Marginal probability density function)

Let $X$ and $Y$ be two Continuous random variables. The marginal distributions can be computed from their joint pdf $p(x,y)$, as a result the marginal pdf of $X$ is $$p_X(x)=\int_\mathscr{Y}p(x,y) \ dy$$ and the marginal pdf of $Y$ is $$p_Y(y)=\int_\mathscr{X}p(x,y) \ dx$$