Marginal Distribution - Two-variable Case

Two-variable Case

Given two random variables X and Y whose joint distribution is known, the marginal distribution of X is simply the probability distribution of X averaging over information about Y. This is 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 Pr(X = x). This is

where Pr(X = x,Y = y) is the joint distribution of X and Y, while Pr(X = x|Y = y) is the conditional distribution of X given Y. In this case, the variable Y has been marginalized out.

Bivariate marginal and joint probabilities for discrete random variables are often displayed as two-way tables.

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. Again, the variable Y has been marginalized out.

Note that a marginal probability can always be written as an expected value:

Intuitively, the marginal probability of X is computed by examining the conditional probability of X given a particular value of Y, and then averaging this conditional probability over the distribution of all values of Y.

This follows from the definition of expected value, i.e. in general