CS 4803A/8803A: Pattern Recognition
Problem Set 1
Date: January 14, 2003 Due: start of class January 23, 2003
1. For each case below, are
and
statistically independent? Why or why not?
(a) The point
is uniformly distributed in the unit square.
(b) The point
is uniformly distributed in the unit circle (a subset of the unit square).
(c) Let
and
be independent and both uniform on
. Let
and
.
(d) Let
and
be independent and both Gaussian with zero mean and unit variance. Let
and
.
(e) Let
and
be independent and both Gaussian with zero mean, but the variance of
is 2
while the variance of
is 1. Again, let
and
.
2. Let
and
be random variables with density functions
! "
$#
&% ')(+*
(
"-,
/.
1032
10
and
"
4
%
5.36.
otherwise.
(a) What is
7"
? Specify the region where the joint density is nonzero.
(b) Using Matlab, generate a plot of
87"
.
(c) Find
and plot that as well. Don’t forget to specify regions of definition.
3. The components
9
and
;:
of a two-dimensional random vector
<
are statistically independent
and have the following marginal densities:
=
>9
%
5.
>9
.
otherwise
and
@?
8:
% A
8:
B.
8:
.
otherwise
C
Let
D
be the event
E9
.
8:
. What are the following?
(a)
F
4<G
(Plot it in 3D in Matlab.)
