The Mathematics of Financial Modelingand Investment Management

6-ConceptsProbability Page 176 Wednesday, February 4, 2004 3:00 PM

176 The Mathematics of Financial Modeling and Investment Management

therefore possible to induce over every set H that belongs to B n a prob-

ability measure, which is the joint probability of the n random variables

Xi. The function

Fx( 1 , ...,xn ) = PX( 1 ≤x 1 , ...,Xn≤xn )

where xi ∈R is called the n-dimensional cumulative distribution func-

tion or simply n-dimensional distribution function (c.d.f. or d.f.). Sup-

pose there exists a function f(x 1 ,...,xn) for which the following relationship

holds:

x 1 xn

Fx( 1 , ...,xn) = ∫...∫fu( 1 , ...,un)du 1 ...dun

• ∞ –∞

The function f(x 1 ,...,xn) is called the n-dimensional probability density

function (p.d.f.) of the random vector X. Given a n-dimensional probabil-

ity density function f(x 1 ,...,xn), if we integrate with respect to all variables

except the j-th variable, we obtain the marginal density of that variable:

∞ ∞

fXj()y = ∫...∫ fu( 1 , ...,un)du 1 ⋅duj – 1 duj + 1 ⋅dun

• ∞ –∞

Given a n-dimensional d.f. we define the marginal distribution func-

tion with respect to the j-th variable, FX()y = PX( j ≤y)as follows:

j

Fxj()y = lim Fx( 1 , ...,xj – 1 , , yxj + 1 , ...,xn)

xi → ∞

ij≠

If the distribution admits a density we can also write

y

FXj()y = ∫ fXj()u ud

• ∞

These definitions can be extended to any number of variables. Given

a n-dimensional p.d.f., if we integrate with respect to k variables

(xi

1

, ...,xi

k

) over Rk, we obtain the marginal density functions with

respect to the remaining variables. Marginal distribution functions with

respect to any subset of variables can be defined taking the infinite limit

with respect to all other variables.