Basic Notions of Probability Theory - Probabilistic Composition of Preferences, Theory and Applications - page 131

A.7.3.2 Multidimensional Normal Distribution

The vector of random variables X= (X 1 , … ,X n ) has a multidimensional normal

2 2

n

distribution with expected values ʼ 1, … , ʼ n , variances r

1 ; ... r

and correlation

coef cients ˁ ij for i ≠ j from 1 to n (X Normal( ʼ , ʣ )) if and only if their joint

*

density has the form

pÞ n det ðRÞ ð 1 = 2 Þ exp f lÞ T R 1 f X x ðÞ ¼½ ð 2 ð 1 = 2 Þð x ð x lÞg;

where ʼ is the vector of coordinates ʼ i and ʣ is the matrix whose entry ʣ ij for i ≠ jis

2
n

the covariance q ij r i r j and for i = j is the variance r .

A.7.3.3 Properties of the Multidimensional Normal Distribution

A T R X Normal ðl;RÞ$ for any matrix A ; ðÞ AX Normal ð A if A Þ:

So,

A T A

X Normal ðl; RÞ for R ¼ ; for a square invertible matrix A $

A 1 ð X lÞ Normal 0 ð ; ; I I denoting the identity matrix ;

and, in particular,

A T A X Normal ðl; RÞ for R ¼ ; for a square invertible matrix A ;!

A − 1 X is a vector of independent coordinates.
The Chi-Squared distribution is the distribution of the sum of squares of normal
distributions.

X

2

Z 2
i

X @

n $

X ¼ for Z 1 ; ... ; Z n i : i : d : Normal 0 ð : ; 1

n

2
n

@

If X , then EX = n,

so that,

X

and X

2

X 2

if X 1 ; ... ; X n are i : i : d : Normal 0 ; r ¼

i ;

then

n

2

E ð X =r Þ ¼ n

and

2

EX ð = n Þ ¼ r :

It is easy to generalize this de nition to:

2
n $

lÞ T R 1 Y @ Y ¼ ð X ð X lÞ