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 .
We want, for all and all , to define and calculate the probability that X ∈ B 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
It is thus natural to say that the conditional density of the random ...