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

Given that X is a real r.v. and Y = (Y1, …, Yn) is a real random vector, we assume that X and Y are independent and that the vector Z = (X,Y1, …, Yn) admits a probability density fZ(x, y1, …, yn).

In this section, we will use as required the notations (Y1, …, Yn) or Y,(y1, …, yn) or y.

Let us recall to begin with .

Conditional probability

We want, for all and all , to define and calculate the probability that XB knowing that Y1 = y1, …, Yn = yn.

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 ...

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