Discrete Stochastic Processes and Optimal Filtering

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:

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

DEFINITION.– Under the hypothesis Ψ X ∈ L¹ (dP), we call mathematical expectation of the random value Ψ X the expression E (Ψ X) defined as:

or, to remind ourselves that X is a vector:

NOTE.– ...