- Suppose has a distribution with
partitioned as and Variance covariance
- Show that if and only if
and are independent. Use my definition of MVN; you are not allowed
to assume that X has a density. A mathematically careful argument my
rely on the fact that if are independent and
are (measurable) functions then
- Show that whether or not is singular, each column of
is in the column space of .
- Show that there is a matrix A which is such that
- Let h be a column of and and be any vectors
such that for i=1,2. Show that .
- Show that if for i=1,2 then
- Show that implies for any .
- Suppose x is such that implies
. Show that the conditional distribution
of given is well-defined and is multivariate
normal with mean
and variance covariance
where a is any solution of
and A is any solution of .
NOTE: Most facts about variates X can be demonstrated
by writing for well chosen A.
- Fix . Minimize and maximize
subject to .
- Fix . Minimize and maximize subject to
for a given symmetric positive definite matrix Q.
- Write out the spectral decomposition of the matrix
and then find a symmetric square root.
- Suppose has a MVN distribution with and
Find the conditional distribution of given and . For which
values of and does this make sense?