The joint MGF of is, for
This can be written
where is the -by- matrix that has 1's everywhere
in the -th column except for 0's in rows
, and has 1's in row , column 1;
row , column 2; etc.
Therefore, from (7.9),
Note that can be a large
-by- matrix if is large. However, if we define
is a reasonable
-by- size matrix and can be easily formed directly.
The final simplified form for the joint MGF is