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

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1.3. Mathematical expectation and applications

1.3.1. Definitions

We are studying a general random vector (not necessarily with a density function):

images

Furthermore, we give ourselves a measurable mapping:

images

Ψ images X (also denoted Ψ (X) or Ψ(X1, …, Xn)) is a measurable mapping (thus an r.v.) defined on (Ω , images).

images

DEFINITION.– Under the hypothesis Ψ images XL1 (dP), we call mathematical expectation of the random value Ψ images X the expression E (Ψ images X) defined as:

images

or, to remind ourselves that X is a vector:

images

NOTE.– ...

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