The Mathematics of Financial Modelingand Investment Management

6-ConceptsProbability Page 194 Wednesday, February 4, 2004 3:00 PM

194 The Mathematics of Financial Modeling and Investment Management

GAUSSIAN VARIABLES

Gaussian random variables are extremely important in probability the-

ory and statistics. Their importance stems from the fact that any phe-

nomenon made up of a large number of independent or weakly

dependent variables has a Gaussian distribution. Gaussian distributions

are also known as normal distributions. The name Gaussian derives

from the German mathematician Gauss who introduced them.

Let’s start with univariate variables. A normal variable is a variable

whose probability distribution function has the following form:

 

1  (x– μ)

(^2) 
fx( μσ,^2 ) = ---------------exp– --------------------
σ 2 π   2 σ^2 
The univariate normal distribution is a distribution characterized by
only two parameters, (μ,σ^2 ), which represent, respectively, the mean and
the variance of the distribution. We write X∼N(μ,σ^2 ) to indicate that
the variable Xhas a normal distribution with parameters (μ,σ^2 ). We
define the standard normal distribution as the normal distribution with
zero mean and unit variance. It can be demonstrated by direct calcula-
tion that if X∼N(μ,σ^2 ) then the variable
Z= X– μ

σ
is standard normal. The variable Z is called the score or Z-score. The
cumulative distribution of a normal variable is generally indicated as

• μ
  Fx()= Φ------------

x

 σ 

where Φ(x) is the cumulative distribution of the standard normal.

It can be demonstrated that the sum of nindependent normal distribu-

tions is another normal distribution whose expected value is the sum of

the expected values of the summands and whose variance is the sum of the

variances of the summands.

The normal distribution has a typical bell-shaped graph symmetrical

around the mean. Exhibit 6.1 shows the graph of a normal distribution.