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
]
- ¹₂
f ()v = [( 2 π)
n
ΣΣΣΣ] exp[(–¹₂)(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