STAT 802

Assignment 1

1. Suppose has a distribution with partitioned as and Variance covariance

1. 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.
2. Show that whether or not is singular, each column of is in the column space of .
3. Show that there is a matrix A which is such that .
4. Let h be a column of and and be any vectors such that for i=1,2. Show that .
5. Show that if for i=1,2 then .
6. Show that implies for any .
7. 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.

2. Fix . Minimize and maximize subject to .
3. Fix . Minimize and maximize subject to for a given symmetric positive definite matrix Q.
4. Write out the spectral decomposition of the matrix

and then find a symmetric square root.

5. Suppose has a MVN distribution with and

Find the conditional distribution of given and . For which values of and does this make sense?

Richard Lockhart
Mon Jan 26 10:30:30 PST 1998