The joint distribution of a pair of variables is a representation of the probability of each combination of results on those two variables. For example, if two variables are each the roll of a die, the distribution would show the probability of each combination of rolls. The marginal distribution is the distribution of one variable, ignoring the other. Sometimes, you have values for the joint distribution, but not the marginals.
Find the probability of each combination of values of the variables. These often will be represented in a table. For example, the rolls of two dice might be represented in a table, with one die on the rows, one on the columns and probabilities written in the cells of the table.
Sum the joint probabilities for each level of the variable for which you are trying to find the marginal distribution. In the example, if you are trying to find the marginal distribution of the die listed on the columns, add up the probabilities for each row of the table. In this case, this will give you six probabilities, each equal to 1/36+1/36+1/36+1/36+1/36+1/36 =1/6.
Write the marginal distribution. In the example, this would be written as: P(x1 = 1) = 1/6 P(x1 = 2) = 1/6 P(x1 = 3) = 1/6 P(x1 = 4) = 1/6 P(x1 = 5) = 1/6 P(x1 = 6) = 1/6
If the variables are continuous, the solution involves integral calculus and will almost always require a computer program such as SAS, R, MATLAB or another statistical package. However, the idea is the same.