on a value less than or equal to and that takes on a value less than or equal to . For random variables , the joint CDF is given by Interpreting the random variables as a random vector yields a shorter notation: Joint density function or mass function Discrete case The joint probability mass function of two discrete random variables is: or written in terms of conditional distributions where is the probability of given that . The generalization of the preceding two-variable case is the joint probability distribution of discrete random variables which is: or equivalently . This identity is known as the chain rule of probability. Since these are probabilities, in the two-variable case which generalizes for discrete random variables to Continuous case The joint probability density function for two continuous random variables is defined as the derivative of the joint cumulative distribution function (see ): This is equal to: where and are the conditional distributions of given and of given respectively, and and are the marginal distributions for and respectively. The definition extends naturally to more than two random variables: Again, since these are probability distributions, one has respectively Mixed case The "mixed joint density" may be defined where one or more random variables are continuous and the other random variables are discrete. With one variable of each type One example of a situation in which one may wish to find the cumulative distribution of one random variable which is continuous and another random variable which is discrete arises when one wishes to use a logistic regression in predicting the probability of a binary outcome Y conditional on the value of a continuously distributed outcome . One must use the "mixed" joint density when finding the cumulative distribution of this binary outcome because the input variables were initially defined in such a way that one could not collectively assign it either a probability density function or a probability mass function. Formally, is the probability density function of with respect to the product measure on the respective supports of and . Either of these two decompositions can then be used to recover the joint cumulative distribution function: The definition generalizes to a mixture of arbitrary numbers of discrete and continuous random variables. Additional properties Joint distribution for independent variables In general two random variables and are independent if and only if the joint cumulative distribution function satisfies Two discrete