The Mathematics of Financial Modelingand Investment Management

6-ConceptsProbability Page 195 Wednesday, February 4, 2004 3:00 PM

Concepts of Probability 195

EXHIBIT 6.1 Graph of a Normal Variable with Zero Mean and σ = 100

Multivariate normal distributions are characterized by the same

exponential functional form. However, a multivariate normal distribu-

tion in nvariables is identified by nmeans, one for each axis, and by a

n×nsymmetrical variance-covariance matrix. For instance, a bivariate

normal distribution is characterized by two expected values, two vari-

ances and one covariance. We can write the general expression of a

bivariate normal distribution as follows:

 1 

exp (^) – ---Q
 2 
fxy( , ) = -----------------------------------------
2 πσXσY 1 – ρ
2
1  x– μX
(^2)  x– μ
X  y– μY  y– μY
(^2) 
Q= --------------- ---------------- – 2 ρ ----------------  --------------- +  --------------- 
1 – ρ^2  σX   σX   σY   σY  
where ρ is the correlation coefficient.