Statistics 851 (Fall 2013)
October 16, 2013
Prof. Michael Kozdron
Lecture #17: Expectation of a Simple Random Variable
Recall that a simple random variable is one that takes on finitely many values.
Definition.
Let (
Ω
,
F
,
P
) be a probability space. A random variable
X
:
Ω
→
R
is called
simple
if it can be written as
X
=
n
i
=1
a
i
1
A
i
where
a
i
∈
R
,
A
i
∈
F
for
i
= 1
,
2
, . . . , n
. We define the expectation of
X
to be
E
(
X
) =
n
i
=1
a
i
P
{
A
i
}
.
Example 17.1.
Consider the probability space (
Ω
,
B
1
,
P
) where
Ω
= [0
,
1],
B
1
denotes the
Borel sets of [0
,
1], and
P
is the uniform probability on
Ω
. Suppose that the random variable
X
:
Ω
→
R
is defined by
X
(
ω
) =
4
i
=1
a
i
1
A
i
(
ω
)
where
a
1
= 4,
a
2
= 2,
a
3
= 1,
a
4
=
−
1, and
A
1
= [0
,
1
2
)
,
A
2
= [
1
4
,
3
4
)
,
A
3
= (
1
2
,
7
8
]
,
A
4
= [
7
8
,
1]
.
Show that there exist finitely many real constants
c
1
, . . . , c
n
and
disjoint
sets
C
1
, . . . , C
n
∈
B
1
such that
X
=
n
i
=1
c
i
1
C
i
.
Solution.
We find
X
(
ω
) =
4
,
if 0
≤
ω
<
1
/
4
,
6
,
if 1
/
4
≤
ω
<
1
/
2
,
2
,
if
ω
= 1
/
2
,
3
,
if 1
/
2
<
ω
<
3
/
4
,
1
,
if 3
/
4
≤
ω
<
7
/
8
,
0
,
if
ω
= 7
/
8
,
−
1
,
if 7
/
8
<
ω
≤
1
,
so that
X
=
7
i
=1
c
i
1
C
i
where
c
1
= 4,
c
2
= 6,
c
3
= 2,
c
4
= 3,
c
5
= 1,
c
6
= 0,
c
7
=
−
1 and
C
1
= [0
,
1
4
)
,
C
2
= [
1
4
,
1
2
)
,
C
3
=
{
1
2
}
,
C
4
= (
1
2
,
3
4
)
,
C
5
= [
3
4
,
7
8
)
,
C
6
=
{
7
8
}
,
C
7
= (
7
8
,
1]
.
17–1