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
)
.
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 Spring '06
 Caire
 Probability theory, WI, valid characteristic function, Φf

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