2.3 Multivariate probability functions

PDFs and PMFs can be functions of more than one rv also.

Example 2.7 (Joint PDF) Consider a PMF of two rvs \(X\) and \(Z\) with a joint probability distribution \[ f_{X,Z}(x, z) = \frac{3(x + z^2)}{5} \quad\text{for $0<x<1$ and $-1<z<1$}. \] This is a valid PMF (check it!), since \[ \int_{-1}^1\int_{0}^1 \frac{3(x + z^2)}{5}\, dx\, dz = 1, \] and \(f_{X,Z}(x,z)\ge0\) for all values of \(x\) and \(z\).

In this example, notice that the sample space is rectangular: in the \(x\)\(z\) plane, the region over which the PMF is non-zero is a rectangle.