The Mathematics of Financial Modelingand Investment Management

6-ConceptsProbability Page 177 Wednesday, February 4, 2004 3:00 PM

Concepts of Probability 177

Any d.f. FXj()y defines a Lebesgue-Stieltjes measure and a Lebesgue-

Stieltjes integral. For example, as we have seen in Chapter 4, in the 1-dimen-

sional case, the measure is defined by the differences FXj()xi – FXj(xi – 1 ).

We can now write expectations in two different, and more useful, ways.

In an earlier section in this chapter, given a probability space (Ω,ℑ,P), we

defined the expectation of a random variable X as the following integral

EX[]= ∫ XPd

Ω

Suppose now that the random variable X has a d.f. FX(u). It can be dem-

onstrated that the following relationship holds:

∞

EX[]= ∫ XPd = ∫ uFd X()u

Ω –∞

where the last integral is intended in the sense of Riemann-Stieltjes. If,

in addition, the d.f. FX

j

()u has a density fX()u = FX ′()u, then we can

write the expectation as follows:

∞ ∞

EX[]= ∫XPd = ∫ uFd X()u = ∫ uf u()ud

Ω –∞ –∞

where the last integral is intended in the sense of Riemann. More in gen-

eral, given a measurable function g the following relationship holds:

∞ ∞

EgX[ ()] = ∫ gu()dFX()u = ∫ gu()fu()ud

• ∞ –∞

This latter expression of expectation is the most widely used in practice.

In general, however, knowledge of the distributions and of distribu-

tion functions of each random variable is not sufficient to determine the

joint probability distribution function. As we will see later in this chap-

ter, the joint distribution is determined by the marginal distributions

plus the copula function.

Two random variables X,Y are said to be independent if

P(X ∈A,Y ∈B) = P(X ∈A)P(Y ∈B)