**1.6. Conditional expectation (concerning random vectors with density function)**

Given that *X* is a real r.v. and *Y* = (*Y*_{1}, …, *Y*_{n}) is a real random vector, we assume that *X* and *Y* are independent and that the vector *Z* = (*X*,*Y*_{1}, …, *Y*_{n}) admits a probability density *f*_{Z}(*x*, *y*_{1}, …, *y*_{n}).

In this section, we will use as required the notations (*Y*_{1}, …, *Y*_{n}) or *Y*,(*y*_{1}, …, *y*_{n}) or *y*.

Let us recall to begin with .

**Conditional probability**

We want, for all and all , to define and calculate the probability that *X* ∈ *B* knowing that *Y*_{1} = *y*_{1}, …, *Y*_{n} = *y*_{n}.

We denote this quantity or more simply . Take note that we cannot, as in the case of discrete variables, write:

The quotient here is indeterminate and equals .

For *j* = 1 at *n*, let us note

We write:

It is thus natural to say that the conditional density of the random ...