Discrete Stochastic Processes and Optimal Filtering

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

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

In this section, we will use as required the notations (Y₁, …, Y_n) or Y,(y₁, …, 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 Y₁ = y₁, …, 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 ...