The joint probability density function \(f\) of two random variables \(X\) and \(Y\) satisfies, for every \(a_1< b_1\) and \(a_2< b_2\), \[ P(a_1\le X\le b_1, a_2\le Y\le b_2) = \int_{a_1}^{b_1} \int_{a_2}^{b_2} f(x,y) dx dy = \int_{a_2}^{b_2} \int_{a_1}^{b_1} f(x,y) dy dx. \]
Unlike for probability mass functions, the probability density function cannot be interpreted directly as a probability. Instead, if we visualize the graph of a pdf as a surface, then we can compute the probability assigned to a rectangle as the volume below the surface (over the rectangle).
Following is an interactive 3-D representation of the graph of a joint density given by \[ f(x,y) = \frac{1}{2\pi} \exp\left(-\frac 12 x^2 - \frac 12 y^2\right), \] which is the probability density function of a two-dimensional standard normal random variable.
The joint (cumulative) distribution function (cdf) \(F\) of \(X\) and \(Y\) is yet another way to summarize the same probabilistic information.
The joint cdf \(F\) is defined through \(F(a,b) = P(X\le a, Y\le b)\) for any real numbers \(a\) and \(b\). Following is an interactive 3-D diagram of this joint cdf \(F\).
The 3-D diagrams on this page are made with JSMol and SageMath.