## 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.