HOMEWORK #1
First Quarter, Course 2010/2011
Due Date: 29/10/2010 (friday, inclass)
c
±
Baltasar Beferull Lozano
General Instructions and Comments
•
The problems are not necessarily given in increasing order of diﬃculty.
•
A ﬁnal result that is correct does not guarantee the maximum grade in an exercise, in the same way
as an incorrect ﬁnal result does not imply a 0 grade in an exercise. You should explain clearly your
arguments and show all the relevant steps in each of the problems. However, please, avoid redundant
verbosity. The grade obtained in a problem will be based mainly in judging what is your level of
understanding when solving the problem, as it is reﬂected from your writing.
•
If any section of your solution of a problem is not legible, that section will be declared null (0 points).
•
You are welcome to ﬂag topics of confusion to you in any problem set; this will not lower your grade.
•
You are expected to do yourself alone all the problems; I will assume so in making up the ﬁnal exam.
Problem 1.
Understanding well Markovity
Suppose the random variables
A
,
B
,
C
,
D
form a Markov chain
A
→
B
→
C
→
D
. Then,
(a) Is
A
→
B
→
C
?
(b) Is
A
→
C
→
D
?
(c) Is
B
→
C
→
D
?
(d) Is
C
→
B
→
A
?
(e) Is
A
→
(
B,C
)
→
D
?
(f) Is
A
→
B
→
(
C,D
) ?
Suppose now that the random variables
A
,
B
,
C
,
D
satisfy that: (i)
A
→
B
→
C
and (ii)
B
→
C
→
D
.
Then, does it follow from these that
A
→
B
→
C
→
D
?
Problem 2.
Inequalities
Label each of the following statements with =,
≤
or
≥
. Label a statement with = if equality always holds.
Label a statement with
≤
or
≥
if strict inequality is possible. Justify each answer.
(a)
H
(
X

Z
) vs.
H
(
X

Y
) +
H
(
Y

Z
)
(b)
h
(
X
+
Y
) vs.
h
(
X
) if
X
and
Y
are independent continuous random variables
(c)
I
(
g
(
X
);
Y
) vs.
I
(
X
;
Y
)
(d)
I
(
Y
;
Z

X
) vs.
I
(
Y
;
Z
) if
p
(
x,y,z
) =
p
(
x
)
p
(
y
)
p
(
z

x,y
)
(e)
I
(
X
1
,X
2
;
Y
1
,Y
2
) vs.
I
(
X
1
;
Y
1
) +
I
(
X
2
;
Y
2
), if
p
(
y
1
,y
2

x
1
,x
2
) =
p
(
y
1

x
1
)
.p
(
y
2

x
2
).
(f)