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Discrete Stochastic Processes and Optimal Filtering by Roger Ceschi, Jean-Claude Bertein

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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 image.

Conditional probability

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

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

images

The quotient here is indeterminate and equals image.

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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