Given two jointly distributed random variables X and Y, the marginal distribution of X is simply the probability distribution of X ignoring information about Y, typically calculated by summing or integrating the joint probability distribution over Y.

For discrete random variables, the marginal probability mass function can be written as P(X=x). This is

where P(X=x,Y=y) is the joint distribution of X and Y, while P(X=x|Y=y) is the conditional distribution of X given Y.

Similarly for continuous random variables, the marginal probability density function can be written as p_X(x). This is

where p_X,Y(x,y) gives the joint distribution of X and Y, while p_X|Y(x|y) gives the conditional distribution for X given Y.

Why the name 'marginal'? One explanation is to imagine the P(X,Y) in a 2D table such as a spreadsheet. The marginals are got by summing the columns (or rows) -- the column sum would then be written in the margin of the table, ie. the column at the side of the table.