The Mathematics of Financial Modelingand Investment Management

6-ConceptsProbability Page 196 Wednesday, February 4, 2004 3:00 PM

196 The Mathematics of Financial Modeling and Investment Management

This expression generalizes to the case of n random variables. Using

matrix notation, the joint normal probability distributions of the random

n vector V = {Xi}, i = 1,2,...,n has the following expression:

V = {}Xi ∼Nn (μμμμ, ΣΣΣ)

where

μi = EX[]i

and ΣΣΣΣis the variance-covariance matrix of the {Xi}

ΣΣΣΣ= E[(V – μμμμ)(V – μμμμ)

T

]

• ¹
  ₂
  f ()v = [( 2 π)
  n
  ΣΣΣΣ] exp[(–¹
  ₂)(v – μμμμ)
  T
  ΣΣΣΣ


• 1
  (v – μμμμ)]

where ΣΣΣΣ = detΣΣΣΣ, the determinant of ΣΣΣΣ.

For n = 2 we find the previous expression for bivariate normal, tak-

ing into account that variances and correlation coefficients have the fol-

lowing relationship

σij = ρijσiσj

It can be demonstrated that a linear combination

n

W = ∑αiXi

i = 1

of n jointly normal random variables Xi ∼N(μi, σi^2 ) with cov(Xi,Xj) =

σij is a normal random variable WN∼ (μW, σW^2 )where

n

μW= ∑αiμi

i = 1

n n

σ^2

W = ∑ ∑αiαjσij

i = 1 j = 1