# 4

# Vector valued random variables

## 4.1 Joint distribution

Let *X* and *Y* be two random variables. The distribution vector does not bear enough information about the relations occurring between *X* and *Y*, for instance to compute where *A* and *B* are generated by *X* and *Y*, respectively; one needs to introduce a new object called the *joint* (*probability*) *distribution* of *X* and *Y*.

**Example 4.1 (Discrete random variables)** Let *X* and *Y* be two discrete random variables on the same probability space . Let , and let (*p*_{1}, . . ., *p*_{n}) and (*q*_{1}, . . ., *q*_{m}) be the mass density vectors of and , respectively.

The image of the map (*X, Y*) : Ω → is a subset of {(*x*_{i}, *y*_{j}), *i* = 1, . . ., *n*, *j* = 1, . . ., *m*} (whose cardinality ...