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 ±nitely many values.
DeFnition.
Let (Ω
,
F
,
P
)beaprobab
i
l
ityspace
. Arandomvar
iab
le
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
.W
ede±netheexpe
c
ta
t
iono
f
X
to be
E
(
X
)=
n
°
i
=1
a
i
P
{
A
i
}
.
Example 17.1.
Consider the probability space (Ω
,
B
1
,
P
)whereΩ=[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
→
R
is de±ned by
X
(
ω
4
°
i
=1
a
i
1
A
i
(
ω
)
where
a
1
=4
,
a
2
=2
,
a
3
,
a
4
=
−
1, and
A
1
=[0
,
1
2
)
,A
2
=[
1
4
,
3
4
)
3
=(
1
2
,
7
8
]
4
7
8
,
1]
.
Show that there exist ±nitely 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 ±nd
X
(
ω
4
,
if 0
≤
ω<
1
/
4
,
6
,
if 1
/
4
≤
1
/
2
,
2
,
if
ω
/
2
,
3
,
if 1
/
2
<ω<
3
/
4
,
1
,
if 3
/
4
≤
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
,
c
2
=6
,
c
3
,
c
4
=3
,
c
5
,
c
6
=0
,
c
7
=
−
1and
C
1
,
1
4
)
,C
2
1
4
,
1
2
)
3
=
{
1
2
}
4
1
2
,
3
4
)
5
3
4
,
7
8
)
6
=
{
7
8
}
7
7
8
,
1]
.
17–1