Page 177 - DMTH404_STATISTICS

Unit 13: Moment Generating Function - Continue

with hi > 0 for all i = ,...,K, then we say that X possesses a joint moment generating function (or Notes
joint mgf) and the function M : H   defined by
X
M (t) = E[exp(t X)]
T
X
is called the joint moment generating function of X.
The following example shows how the joint moment generating function of a standard
multivariate normal random vector is calculated:

Example: Let X be a K × 1 standard multivariate normal random vector. Its support R is
X
R =  K
X
and its joint probability density function f (x) is
X
T 

f (x) (2 )  K /2 exp  1 x x 



X
 2 
Therefore, the joint moment generating function of X can be derived as follows:
M (t) = E[exp(t X)]
T
X
= E[exp(t X + t X + ... + t X )]
1 1 2 2 K K
 K 
= E  exp(t X )
i 
i
 i 1 

K
=   E exp(t X )  (by mutual indepdence of the entries of X)
i
i

i 1
K
=  M (t ) (by the definition of moment generating function)
X
i
i 1 i

where we have used the fact that the entries of X are mutually independent (see mutual
independence via expectations) and the definition of the moment generating function of a
random variable. Since the moment generating function of a standard normal random variable
is
 1 2 
M (t ) = exp  t i 
X i i  2 
then the joint moment generating function of X is
K
MX(t) =  M (t )
i
X
i

i 1
K 1
=  exp   t 2 
i 
i 1  2 

 1 K 2 
= exp   t i 
 2 i 1 

 1 T 
= exp  t t 
 2 

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