Marginal density in the context of Joint distribution

Given random variables ${\displaystyle X,Y,\ldots }$, that are defined on the same probability space, the multivariate or joint probability distribution for ${\displaystyle X,Y,\ldots }$ is a probability distribution that gives the probability that each of ${\displaystyle X,Y,\ldots }$ falls in any particular range or discrete set of values specified for that variable. In the case of only two random variables, this is called a bivariate distribution, but the concept generalizes to any number of random variables.

The joint probability distribution can be expressed in terms of a joint cumulative distribution function and either in terms of a joint probability density function (in the case of continuous variables) or joint probability mass function (in the case of discrete variables). These in turn can be used to find two other types of distributions: the marginal distribution giving the probabilities for any one of the variables with no reference to any specific ranges of values for the other variables, and the conditional probability distribution giving the probabilities for any subset of the variables conditional on particular values of the remaining variables.