where we used the usual notations for the determinant, transpose, and inverse of a matrix. The vector of means ![mu]()
may have any elements in ![double-struck upper R]()
, but, just as in the one‐dimensional case, the standard deviation has to be positive. In the multivariate case, the covariance matrix ![sigma-summation]()
has to be symmetric and positive definite.
The multivariate normal defined thus has many nice properties. The basic one is that the one‐dimensional distributions are all normal, that is, ![upper X Subscript i Baseline tilde upper N left-parenthesis mu Subscript i Baseline comma sigma Subscript i i Baseline right-parenthesis]()
and ![Cov left-parenthesis upper X Subscript i Baseline comma upper X Subscript j Baseline right-parenthesis equals sigma Subscript i j Baseline]()
. This is also true for any marginal. For example, if ![left-parenthesis upper X Subscript r Baseline comma ellipsis comma upper X Subscript k Baseline right-parenthesis]()
are the last coordinates, then
![Start 4 By 1 Matrix 1st Row upper X Subscript r Baseline 2nd Row upper X Subscript r plus 1 Baseline 3rd Row vertical-ellipsis 4th Row upper X Subscript k Baseline EndMatrix tilde upper M upper V upper N Subscript k minus r plus 1 Baseline left-parenthesis Start 4 By 1 Matrix 1st Row mu Subscript r Baseline 2nd Row mu Subscript r plus 1 Baseline 3rd Row vertical-ellipsis 4th Row mu Subscript k Baseline EndMatrix comma Start 4 By 4 Matrix 1st Row 1st Column sigma Subscript r comma r Baseline 2nd Column sigma Subscript r comma r plus 1 Baseline 3rd Column ellipsis 4th Column sigma Subscript r comma k Baseline 2nd Row 1st Column sigma Subscript r plus 1 comma r Baseline 2nd Column sigma Subscript r plus 1 comma r plus 1 Baseline 3rd Column ellipsis 4th Column sigma Subscript r plus 1 comma k Baseline 3rd Row 1st Column vertical-ellipsis 2nd Column vertical-ellipsis 3rd Column down-right-diagonal-ellipsis 4th Column vertical-ellipsis 4th Row 1st Column sigma Subscript k comma r Baseline 2nd Column sigma Subscript k comma r plus 1 Baseline 3rd Column ellipsis 4th Column sigma Subscript k comma k Baseline EndMatrix right-parenthesis period]()
So any particular vector of components is normal.
Conditional distribution of a multivariate normal is also a multivariate normal. Given that ![bold upper X]()
is a ![upper M upper V upper N Subscript k Baseline left-parenthesis mu comma sigma-summation right-parenthesis]()
and using the vector notations above assuming that ![bold upper X 1 equals left-parenthesis upper X 1 comma ellipsis comma upper X Subscript r Baseline right-parenthesis]()
and ![bold upper X 2 equals left-parenthesis upper X Subscript r plus 1 Baseline comma ellipsis comma upper X Subscript k Baseline right-parenthesis]()
, then we can write the vector ![mu]()
and matrix ![sigma-summation]()
as
![mu equals StartBinomialOrMatrix mu 1 Choose mu 2 EndBinomialOrMatrix and sigma-summation equals Start 2 By 2 Matrix 1st Row 1st Column sigma-summation Underscript 11 Endscripts 2nd Column sigma-summation Underscript 12 Endscripts 2nd Row 1st Column sigma-summation Underscript 21 Endscripts 2nd Column sigma-summation Underscript 22 Endscripts EndMatrix comma]()
where the dimensions are accordingly chosen to match the two vectors (![r]()
and ![k minus r]()
). Thus, the conditional distribution of ![bold upper X 1]()
given ![bold upper X 2 equals bold a]()
, for some vector ![bold a]()
is
![bold upper X 1 vertical-bar bold upper X 2 equals bold a tilde upper M upper V upper N Subscript r Baseline left-parenthesis mu 1 minus sigma-summation Underscript 12 Endscripts sigma-summation Underscript 22 Overscript negative 1 Endscripts left-parenthesis mu 2 minus bold a right-parenthesis comma sigma-summation Underscript 11 Endscripts minus sigma-summation Underscript 12 Endscripts sigma-summation Underscript 22 Overscript negative 1 Endscripts sigma-summation Underscript 21 Endscripts right-parenthesis period]()
Furthermore, the vectors ![bold upper X 2]()
and ![bold upper X 1 minus sigma-summation Underscript 21 Endscripts sigma-summation Underscript 22 Overscript negative 1 Endscripts bold upper X 2]()
are independent. Finally, any affine transformation ![upper A upper X plus b]()
, where ![upper A]()
is a ![k times k]()
matrix and ![b]()
is a ![k]()
‐dimensional constant vector, is also a multivariate normal with mean vector ![upper A mu plus b]()
and covariance matrix ![upper A sigma-summation
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