Given a random pair (or a random vector), the marginal probability distribution models the distribution of one of (or a subset of) the variables.
For a discrete random pair, the marginal distribution of a variable gives the probability of observing its values independently of the value of the other. It can be computed as the sum of the probabilities of observing pair of values for all possible values of the second variable.
For a continuous random pair, the marginal probability density function of a variable is obtained by summing (integrating) the joint probability density function over all possible values of the second variable.
Given a discrete random pair $(X,Y)\in\X\times\Y$, the marginal probability distribution of $X$ (respectively $Y$) is defined as $$ P_{X} : \Sigma_X \rightarrow [0,1] $$ with $$ P_{X} ( x ) = P(X=x ) = \sum_{y\in\Y} P(X=x,\ Y=y) = \sum_{y\in\Y} P_{X,Y}(x,\ y) , $$ where $P_{X,Y}$ is the joint probability distribution of $(X,Y)$.
Given a continuous random pair $(X,Y)\in\X\times\Y$, the marginal probability density function of $X$ is defined as $$ p_{X} : \X \rightarrow \R^+ $$ with $$ p_{X}( x ) = \int_{\Y} p_{X,Y}(x,y) dy $$ where $p_{X,Y}$ is the joint probability density function of $(X,Y)$.