Assignment 1

- 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 are independent. - 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 meanand 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*. - Show that if and only if
and are independent. Use my definition of MVN; you are not allowed
to assume that
- 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?

Mon Jan 26 10:30:30 PST 1998