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, …, Xn)) 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: