The Mathematics of Financial Modelingand Investment Management

6-ConceptsProbability Page 192 Wednesday, February 4, 2004 3:00 PM

192 The Mathematics of Financial Modeling and Investment Management

sum of the values taken by the two variables X,Y. Let’s suppose that the

two variables X(ω), Y(ω) have a joint density p(x,y) and marginal densi-

ties pX(x) and pY(x), respectively. Let’s call H the cumulative distribu-

tion of the variable Z. The following relationship holds

Hu()= PZ [ ()ω ≤u]= ∫ ∫ px y ( , )dyxd

A

A = {y ≤–x + u}

In other words, the probability that the sum X + Y be less than or equal

to a real number u is given by the integral of the joint probability distri-

bution function in the region A. The region A can be described as the

region of the x,y plane below the straight line y = –x + u.

If we assume that the two variables are independent, then the distri-

bution of the sum admits a simple representation. In fact, under the

assumption of independence, the joint density is the product of the mar-

ginal densities: p(x,y) = pX(x)pY(x). Therefore, we can write

∞



uy–



Hu()= PZ [ ()ω ≤u]= ∫ ∫px y ( , )dyxd = ∫ ()xd pY y



 ∫ pX x ()yd

A –∞ –∞ 

We can now use a property of integrals called the Leibnitz rule,

which allows one to write the following relationship:

∞

dH

-------- = pZ ()u = ∫pX(uy– )pY ()y yd

du –∞

Recall from Chapter 4 that the above formula is a convolution of

the two marginal distributions. This formula can be reiterated for any

number of summands: the density of the sum of n random variables is

the convolution of their densities.

Computing directly the convolution of a number of functions might

be very difficult or impossible. However, if we take the Fourier transforms

of the densities, PZ(s), PX(s), PY(s) computations are substantially simpli-

fied as the transform of the convolution is the product of the transforms:

∞

pZ ()u = ∫ pX(uy– )pY ()y yd ⇒PZ ()s = PX ()s ×PY ()s

• ∞