## Mathematical Expectation

With these new integrals introduced, we can now answer the second question stated at the end of the introduction: How do we define the mathematical expectation if the distribution of X is neither discrete nor absolutely continuous?

Definition 2.12: The mathematical expectation of a random variable X is defined as E(X) = f X(o)dP(o) or equivalently as E(X) = f xdF(x) (cf(2.15)), whereFis the distribution function ofX, provided that the integrals involved are defined. Similarly, if g(x) is a Borel-measurable function on Kk and Xis a random vector in Kk, then, equivalently, E[g(X)] = f g(X(o))dP(o) = f g(x )dF(x), provided that the integrals involved are defined.

Note that the latter part of Definition 2.12 covers both examples (2.1) and (2.3).

As motivated in the introduction, the mathemat...

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