p. 7-1
Expectation
• Recall
. Expectation for univariate random variable.
• Theorem. For random variables
X
=(
X
1
, … ,
X
n
) with joint pmf
p
X
/pdf
f
X
, the
expectation
of a univariate random variable
Y
, where
Y
=
g
(
X
1
, … ,
X
n
),
g
:
R
n
→
R
1
,
is
if
X
1
, … ,
X
n
are discrete and the sum converges absolutely, or
if
Y
and
X
1
, … ,
X
n
are continuous and the integrals converges
absolutely.
Proof
. Like the univariate case.
Q
: What if
Y
is discrete and
X
1
, … ,
X
n
are continuous?
E
(
Y
)
≡
y
∈
Y
yp
Y
(
y
)
=
x
=(
x
1
,...,x
n
)
∈
X
g
(
x
1
,...,x
n
)
p
X
(
x
1
n
)
≡
E
[
g
(
X
1
,...,X
n
)]
(1)
(2)
(3)
(4)
E
(
Y
)
≡
∞
−∞
yf
Y
(
y
)
dy
=
∞
−∞
···
∞
−∞
g
(
x
1
n
)
f
X
(
x
1
n
)
dx
1
dx
n
≡
E
[
g
(
X
1
n
)]
p. 7-2
Notation.
Shorthand notation. Combine (1) and (3), by writing
Riemann-Stieltjes Integral. For example, for non-negative
g
,
and combine (2) and (4) by writing
where the limit is taken over all
a
=
x
0
<
x
1
<
<
x
n
=
b
as
n
→
0
and
[Recall
. The integral of
g
over (a, b]
is defined as
max
i
=1
,...,n
(
x
i
−
x
i
−
1
)
→
0
.
b
a
g
(
x
)
dF
(
x
)=lim
n
i
=1
g
(
x
i
)[
F
(
x
i
)
−
F
(
x
i
−
1
)]
.
b
a
g
(
x
)
dx
=lim
n
i
=1
g
(
x
i
)(
x
i
−
x
i
−
1
)
.
]
Note.
g
(
X
1
, … ,
X
n
)=
X
i
⇒
E
[
g
(
X
1
, … ,
X
n
)]=
E
(
X
i
)
g
(
X
1
, … ,
X
n
)=(
X
i
a aa
)
2
⇒
E
[
g
(
X
1
, … ,
X
n
)]=
Var
(
X
i
)
μ
X
i
.
μ
X
i
σ
2
X
i
.