For two or more continuous random variables, joint distributions are described by a joint CDF F(x,y) (applicable to any type of random variable) and, in the continuous case, a joint PDF obtained via mixed partial differentiation of the joint CDF, from which probabilities are computed by integrating the density over a region; marginal distributions are recovered by integrating the joint PDF over the other variable(s), and conditional distributions are defined analogously to conditional probability as the joint density divided by the relevant marginal density, satisfying a continuous analogue of Bayes' rule. Independence of continuous random variables is formally defined as factorization of the joint PDF (or equivalently the joint CDF) into the product of the marginals for all values, and the two-dimensional law of the unconscious statistician (2D LOTUS) extends expectation computation to functions of multiple jointly distributed random variables without deriving the distribution of the function itself, yielding as a key consequence that independence implies E(XY) = E(X)E(Y) (independence implies zero correlation). This belongs to probability theory, specifically multivariate distribution theory, extending single-variable concepts of CDF, PDF, conditioning, and LOTUS to the joint treatment of multiple random variables.
For two or more continuous random variables, joint distributions are described by a joint CDF F(x,y) (applicable to any type of random variable) and, in the continuous case, a joint PDF obtained via …