STAT 350
Assignment 2: Solutions
is a density.
. Show that
HINT: What is 
in terms of I.
Solution
We have
do the double integral J in polar co-ordinates ( 
,
) to show J=1.
Solution
When you do an integral in polar co-ordinates you have to:
replace each x in the integrand with 
and each y
with 
, replace dxdy with 
,
and find the set of 
values which correspond to the set
of x,y values over which we are integrating. The Jacobian is the
absolute value of the determinant filled up with derivatives of (x,y)
with respect to r and 
. This 2 by 2 matrix has determinant
r. The value of r, being a distance from the origin is in the
range 0 to 
while the angle in the plane is measured over
any interval of length 
such as 
. This makes
The integral gives leaving
is a density.
Solution
All you have to do is prove that 
and 
.
But is clearly positive. Thus I>0 and since 
we have I=1.
are independent 
random
variables, so that 
with 
independent
standard normals.
and 
express X in the
form AZ+b for a suitable matrix A and vector b.
Solution
We have 
and 
.
and identify 
and 
.
Solution
The definition of MVN is that X be of the form AZ+b and then
and 
. So 
and
.
for i=1,2,3 and 
. Show
that 
and find 
and 
.
Solution
Let B be the matrix
Then Y=BX so Y is 
.
Arithmetically we find
and
where 
has 
columns.
as a partitioned (3 rows, 3 columns) matrix.
Solution
is called block diagonal. Show that 
exists
if and only if each 
exists and that then
is block diagonal.
Solution
Check by multiplying that
This shows that if each 
is invertible then so is A. To
do the converse suppose that B is and partition B
into a 
array with entries 
. Multiply AB and
set this equal to the identity. You get 9 equations like
and 
. The first such
equation shows that 
must be invertible and that 
must
be the inverse of . The second equation then shows (because
we now know that A-1 is invertible that 
.
for i=1,2 and 
.
Show that is block diagonal and give a formula for
.
Solutions
The conditions show that all the off-diagonal blocks are 0, remembering
that 
. Thus
is partitioned
to conform with the partitioning of X (that is 
is a scalar
and 
is a column vector of length for i=1,2.
Let 
be obtained by fitting
by least square, 
be obtained by fitting
and similarly for 
. Let 
be the usual
least squares estimate for
Show that 
.
Solutions
Multiply out the partitioned matrix 
to
get
be the vectors of fitted values corresponding to the
estimates 
for i=1,2,3. Show that for 
we have
.
Solution
because the centre term .
of the first assignment identify
and 
and verify the orthogonality condition of this
problem.
Solution
In the solution set for assignment the first column of
is 1, the second is and the other two columns are .
Now just multiply things like 
to make sure you get 0.
Solution