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
1.3: Mathematical expectation and applications
1.3 Mathematical expectation and applications
1.3.1 Definitions
We are studying a general random vector (not necessarily with a density function):
Furthermore, we give ourselves a measurable mapping:
? ? X (also denoted ? ( X) or ?( X 1, , X n)) is a measurable mapping (thus an r.v.) defined on ( ?, a).
DEFINITION. Under the hypothesis ? ? X ? L 1 ( dP), we call mathematical expectation of the random value ? ? X the expression E( ? ? X) defined as:
or, to remind ourselves that X is a vector:
| Note | This definition of the mathematical expectation of ? ? X is well adapted to general problems or to those of a more theoretical orientation; in particular, it is by using the latter that we construct L 2 ( dP) the Hilbert space of the second order r.v. |
In practice, however, it is the P X law (similar to the measure P by the mapping X) and not P that we do not know. We thus want to use the law P X to express E( ? ? X), and it is said that the calculation of E( ? ? X) from the space ( ?, a, P) to the space 
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In order to simplify the writing in the theorem that follows (and as will often occur...
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