The Mathematics of Financial Modelingand Investment Management

6-ConceptsProbability Page 175 Wednesday, February 4, 2004 3:00 PM

Concepts of Probability 175

tion defined over the set of outcomes. Therefore, the inverse image of A,

X–1(A) belongs to ℑ and has a well-defined probability P(X–1(A)).

The measure P thus induces another measure on the real axis called

distribution or distribution law of the random variable X given by:

μX(A) = P(X–1(A)). It is easy to see that this measure is a probability

measure on the Borel sets. A random variable therefore transfers the

probability originally defined over the space Ω to the set of real numbers.

The function F defined by: F(x) = P(X ≤ x) for x ∈ R is the cumula-

tive distribution function (c.d.f.), or simply distribution function (d.f.),

of the random variable X. Suppose that there is a function f such that

x

Fx() = ∫ fyd

• ∞

or F′(x) = f(x), then the function f is called the probability density func-

tion of the random variable X.

RANDOM VECTORS

After considering a single random variable, the next step is to consider

not only one but a set of random variables referred to as random vectors.

Random vectors are formed by n-tuples of random variables. Consider a

probability space (Ω,ℑ,P). A random variable is a measurable function

from Ω to R^1 ; a random vector is a measurable function from Ω to Rn.

We can therefore write a random vector X as a vector-valued function

f(ω) = [f 1 (ω) f 2 (ω) ... fn(ω)]

Measurability is defined with respect to the Borel σ-algebra B n. It can

be demonstrated that the function f is measurable ℑ if and only if each

component function fi(ω) is measurable ℑ.

Conceptually, the key issue is to define joint probabilities (i.e., the

probabilities that the n variables are in a given set). For example, con-

sider the joint probability that the inflation rate is in a given interval

and the economic growth rate in another given interval.

Consider the Borel σ-algebra B n on the real n-dimensional space Rn.

It can be demonstrated that a random vector formed by n random vari-

ables Xi, i = 1,2,...,n induces a probability measure over B n. In fact, the

set (ω ∈ Ω: (X 1 (ω),X 2 (ω),...,Xn(ω)) ∈ H; H ∈ B n) ∈ ℑ (i.e., the inverse

image of every set of the σ-algebra B n belongs to the σ-algebra ℑ). It is