STAT 350
Assignment 2: Solutions
is a density.
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
Solution
All you have to do is prove that and
.
But
is clearly positive. Thus I>0 and since
we have I=1.
Solution
We have and
.
Solution
The definition of MVN is that X be of the form AZ+b and then
and
. So
and
.
Solution
Let B be the matrix
Then Y=BX so Y is .
Arithmetically we find
and
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
.
Solutions
The conditions show that all the off-diagonal blocks are 0, remembering
that . Thus
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
Solution
because the centre term .
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