Discrete Stochastic Processes and Optimal Filtering
Concerned with the founding principles of optimal filters, this text presents several reminders about both random vectors and Gaussian vectors, and allows readers to tackle digital filtering.
TABLE OF CONTENTS Discrete Stochastic Processes and Optimal Filtering
1.6: Conditional expectation (concerning random vectors with density function)
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 P(( X ? B)( Y 1 = y 1) ? ?( Y n= y n)) or more simply P ( X ? B y 1, , y n). 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 I j = [ y j, y j+h[
We...
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