This preview shows pages 1–2. Sign up to view the full content.
EE 464
Mitra, Spring 2005
Final Solutions
TOTAL = 93
MEAN = 46.22
MAX = 74
MIN= 17
1.
Problem 1
Basics
(a) Give a defnition oF a probability space. Give an example oF a fnitedimensional probability
space and oF an infnitedimensional probability space.
A probability space is defned by the triple:
(Ω
,
F
,
P
)
where
Ω
is the sample space,
F
is the
associated feld and
P
is the probability measure. An example oF a fnite dimensional probability
space is the Fair coin example with:
Ω =
{
H,T
}
,
F
=
{
φ,
{
H,T
}
,
{
H
}
,
{
T
}}
= the power
set and the probability distribution given by
P
[
H
] =
P
[
T
] =
1
2
. An example oF an infnite
dimensional probability space is a uniForm type distribution, that is,
Ω = [0
,
1]
,
F
=
B
[0
,
1]
=
the Borel sets For
[0
,
1]
, and For any subinterval oF
[0
,
1]
,
P
[
a,b
] =
b
−
a
.
(b) Give a defnition oF a random variable.
A random variable is defned on a valid probability space. It is a mapping
X
: Ω
→
R
(a
mapping From the sample space to the real line) such that the set
{
X
≤
x
}
is a valid event (
i.e.
an element oF
F
) and
P
[
X
=
∞
] =
P
[
X
=
−∞
] = 0
.
2.
Problem 2
Characteristic ±unctions
(a) Show that iF
Φ(
ω
)
is a valid characteristic Function then

Φ(
ω
)

2
is also a valid characteristic
Function. Can you describe how such a characteristic Function might come about?

Φ(
ω
)

2
=
Φ(
ω
)Φ(
ω
)
∗
=
E
[
e
jωX
]
E
[
e
jωX
]
∗
=
E
[
e
jωX
]
E
[
e
−
jωX
]
Thus

Φ(
ω
)

2
is the characteristic Function oF
X
1
−
X
2
where
X
1
and
X
2
are independent and
identically distributed random variables. Clearly iF
X
1
and
X
2
are valid random variables, so is
the sum and thus

Φ(
ω
)

2
is a valid characteristic Function.
(b) Let
f
(
x
)
and
g
(
x
)
be two valid probability density Functions. ±orm the mixture probability
density Function
h
(
x
) =
ǫf
(
x
) + (1
−
ǫ
)
g
(
x
)
where
0
< ǫ <
1
. Determine the characteristic
Function oF a random variable
X
with pdF
h
(
x
)
.
Φ(
ω
)
=
i
∞
−∞
h
(
x
)
e
jωx
dx
=
i
∞
−∞
{
ǫf
(
x
) + (1
−
ǫ
)
g
(
x
)
}
e
jωx
dx
=
ǫ
i
∞
−∞
f
(
x
)
e
jωx
dx
+ (1
−
ǫ
)
i
∞
−∞
g
(
x
)
e
jωx
dx
=
ǫ
Φ
f
(
ω
) + (1
−
ǫ
)Φ
g
(
ω
)
where
Φ
f
(
ω
)
is the characteristic Function oF a random variable with pdF
f
(
x
)
and
Φ
g
(
ω
)
is the
characteristic Function oF a random variable with pdF
g
(
x
)
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '06
 Caire

Click to edit the document details