Multivariate Probability Density

If we have several continuous variables $x_1,\dots ,x_D$ denoted collectively by the vector $\mathbf{x}$, then we can define a joint probability density

$$p(\mathbf{x})=p(x_1,\dots ,x_D)$$

such that the probability of $\mathbf{x}$ falling in an infinitesimal volume $\delta \mathbf{x}$ containing the point $\mathbf{x}$ is given by $p(\mathbf{x})\delta \mathbf{x}$.

This multivariate PDF must satisfy:

$$\begin{array}{r}p(\mathbf{x})\ge 0\\ \int p(\mathbf{x})dx=1\end{array}$$

in which the integral is taken over the whole $\mathbf{x}$ space.

More generally, we can also consider joint probability distributions over a combination of discrete and continuous variables.