ECE 302, Homework #6, Due 2/25
http://www.ece.purdue.edu/
∼
chihw/ECE302
09S.html
Review of Calclulus: The chain rule.
Question 1:
Consider the following functions.
F
(
x
)
=
Z
x
0
f
(
s
)
ds
g
(
x
)
=
F
(
h
(
x
)) =
Z
h
(
x
)
0
f
(
s
)
ds
Express
d
dx
F
(
x
) and
d
dx
g
(
x
) using
f
(
x
) and
h
(
x
). (Hint: You should use the chain rule
that
d
dx
g
(
x
) =
d
ds
F
(
s
)
fl
fl
s
=
h
(
x
)
×
d
dx
h
(
x
).)
Question 2:
[Intermediate/Exam Level] (Compare it with HW4Q1). Consider a continu
ous random variable
X
with the following pdf
f
X
(
x
):
f
X
(
x
) = 1
.
5
e

3

x

for all
x
(1)
Consider a discrete “quantizer”
Y
of the magnitude of
X
as follows.
For any
X
, if
k
≤ 
X

< k
+ 1, then
Y
=
k
.
For example, if the
X
value is

π
, then
Y
= 3 since
3
≤  
π

<
4. If the
X
value is 1
.
25, then
Y
= 1. Find the pmf of the discrete variable
Y
. Namely, find
P
(
Y
=
k
) for
k
= 0
,
1
,
2
· · ·
. What type of random variables is
Y
?
Question 3:
[Intermediate/Exam Level]
Let
X
=
Y
1
+
Y
2
+
· · ·
+
Y
n
be a binomial random variable that results from the
summation of
n
independent Bernoulli random variables
Y
1
to
Y
n
. The success probability
of
Y
i
is
p
for
i
= 1
,
· · ·
, n
.
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 Spring '08
 GELFAND
 Probability theory, CDF, Intermediate/Exam Level

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