Basic properties of expectations of discrete random
variables
Property 1 (Absolute integrability): Suppose ξ is a
discrete random variable. Then Eξ is finite if and
only if E|ξ| < ∞. Further
Eξ = Eξ + − Eξ − ,
E|ξ| = Eξ + + Eξ − .
Property 2 (Linearity): Suppose ξ and η are discrete
random variables. If Eξ and Eη exist, then
E(aξ + bη) = aEξ + bEη.
Property 3 (Monotonicity): Suppose ξ and η are
discrete random variables. If ξ ≤ η and the
expectations of ξ and η exits, then Eξ ≤ Eη.
Property 4 (Modulus inequality): Suppose ξ and η
are discrete random variables. If |ξ| ≤ η and the
expectation Eη exists, then Eξ exists and
|Eξ| ≤ E|ξ| ≤ Eη.
Property 4 (Modulus inequality): Suppose ξ and η
are discrete random variables. If |ξ| ≤ η and the
expectation Eη exists, then Eξ exists and
|Eξ| ≤ E|ξ| ≤ Eη.
P
P
Proof. Write ξ = i xi IAi , η = j yj IBj , where
xi , yj ≥ 0.
Property 4 (Modulus inequality): Suppose ξ and η
are discrete random variables. If |ξ| ≤ η and the
expectation Eη exists, then Eξ exists and
|Eξ| ≤ E|ξ| ≤ Eη.
P
P
Proof. Write ξ = i xi IAi , η = j yj IBj , where
xi , yj ≥ 0. Then
X
X
yj IAi Bj , |ξ| =
|xi |IAi Bj .
η=
i,j
i,j
So on the event Ai Bj , yj ≥ |xi |.
Property 4 (Modulus inequality): Suppose ξ and η
are discrete random variables. If |ξ| ≤ η and the
expectation Eη exists, then Eξ exists and
|Eξ| ≤ E|ξ| ≤ Eη.
P
P
Proof. Write ξ = i xi IAi , η = j yj IBj , where
xi , yj ≥ 0. Then
X
X
yj IAi Bj , |ξ| =
|xi |IAi Bj .
η=
i,j
i,j
So on the event Ai Bj , yj ≥ |xi |. Therefore,
yj IAi Bj ≥ |xi |IAi Bj . By Property 3,
yj P (Ai Bj ) ≥ |xi |P (Ai Bj ).
so
∞ > Eη =
X
≥
X
yj P (Bj ) =
X
j
yj P (Ai Bj )
i,j
|xi |P (Ai Bj ) =
i,j
X
|xi |P (Ai ).
i
Hence Eξ exists and
X
X
|xi |P (Ai )
xi P (Ai )| ≤
|Eξ| =|
i
=E|ξ| ≤ Eη < ∞.
i
Property 5 (Monotone convergence theorem):
Suppose ξn and ηn are discrete random variables. If
0 ≤ ξn %, 0 ≤ ηn % and
lim ξn = lim ηn = ξ,
n
n
(ξ is not necessarily discrete), then
lim Eξn = lim Eηn .
n
n
In particular, if 0 ≤ ξn % ξ and ξ is also a discrete
r.v., then Eξn % Eξ.
Proof. It is sufficient to show that for any m,
lim Eξn ≥ Eηm .
n
So it is sufficient to show that, if
0 ≤ ξn % ξ ≥ η
and η is a discrete random variable, then
lim Eξn ≥ Eη.
n
Notice, for any discrete random variable
P
η = i yi IAi ≥ 0, we have
η≥
l
X
i=1
yi IAi
l
X
and E[
yi IAi ] → Eη.
i=1
Without loss of generality, we can assume η is a
simple random variable.
Write An = {ξn < η − }, then An & ∅,
ξn =ξn IAn + ξn IACn
≥(η − )IACn ≥ η − − (η − )IAn
≥η − − κIAn ,
where κ is the maximum value of η.
Write An = {ξn < η − }, then An & ∅,
ξn =ξn IAn + ξn IACn
≥(η − )IACn ≥ η − − (η − )IAn
≥η − − κIAn ,
where κ is the maximum value of η.
It follows that
Eξn ≥ Eη − − κP (An ) → Eη − .
The proof is completed.
Properties of Mathematical expectation for general
random variables
For a random variable ξ. Define
ξ (m) =
k
2m
if
k
k+1
<ξ≤ m .
m
2
2
Then
If ξ ≥ 0, then 0 ≤ ξm % ξ and
0 ≤ ξ − ξ (m) ≤ 21m .
In general, |ξ − ξ (m) | ≤
1
2m .
Theorem
Eξ exists if and only if Eξ (m) exists for one m
(and then all m). Further,
Eξ = lim Eξ (m) .
m→∞
Suppose ξ has cdf F (x). Write xm,k =
ξ
(m)
=
E|ξ (m) | =
=
=
∞
X
k=−∞
∞
X
k=−∞
∞
X
Then
xm,k I{xm,k < ξ ≤ xm,k+1 },
|xm,k |P (xm,k < ξ ≤ xm,k+1 )
|xm,k |∆F (xm,k )
k=−∞
∞ Z
X
k=−∞
k
2m .
|xm,k |dF (x),
xm,k <x≤xm,k+1
where ∆F (xm,k ) = F (xm,k+1 ) − F (xm,k ).
For xm,k
So,
< x ≤ xm,k+1 , we have |xm,k | − |x| ≤
Z
∞
|x|dF (x) −
≤
=
−∞
∞
X
1
2m
|xm,k |∆F (xm,k )
k=−∞
∞ Z
X
k=−∞
Z ∞
≤
|x|dF (x) +
−∞
|xm,k |dF (x)
xm,k <x≤xm,k+1
1
.
2m
So, Eξ exists if and only if Eξ (m) exits.
1
2m .
Similarly,
Eξ
(m)
=
=
∞
X
xm,k P (xm,k < ξ ≤ xm,k+1 )
k=−∞
∞
X
xm,k ∆F (xm,k )
k=0
and
Z ∞
1
xdF (x) − m ≤ Eξ (m) ≤
2
−∞
The proof is completed.
Z
∞
xdF (x) +
−∞
1
.
2m
Corollary
Suppose ξ ≥ 0 and that {ηn } is a sequence of
discrete random variables with 0 ≤ ηn % ξ. Then
Eξ = lim Eηn .
n→∞
Corollary
Suppose ξ ≥ 0 and that {ηn } is a sequence of
discrete random variables with 0 ≤ ηn % ξ. Then
Eξ = lim Eηn .
n→∞
Proof. Note 0 ≤ ηn % ξ, 0 ≤ ξ (n) % ξ. So
lim Eηn = lim Eξ (n) = Eξ.
n→∞
n→∞
3.1.5 Basic properties of expectations
3.1.5 Basic properties of expectations
3.1.5 Basic properties of expectations
3.1.5 Basic properties of expectations
1
(Positivity). If 0 ≤ ξ, then
Eξ ≥ 0.
3.1.5 Basic properties of expectations
3.1.5 Basic properties of expectations
1
(Positivity). If 0 ≤ ξ, then
Eξ ≥ 0.
2
(Linearity). Eξ and Eη exist =⇒
E(aξ + bη) = aEξ + bEη.
3.1.5 Basic properties of expectations
Proof of Property (2). Note |X − X (m) | ≤ 21m .
So
1 + |a| + |b|
(m)
(m)
(m) .
(aξ + bη) − (aξ + bη ) ≤
2m
3.1.5 Basic properties of expectations
Proof of Property (2). Note |X − X (m) | ≤ 21m .
So
1 + |a| + |b|
(m)
(m)
(m) .
(aξ + bη) − (aξ + bη ) ≤
2m
It follows that
1 + |a| + |b|
(m)
(m)
(m) − (aEξ + bEη ) ≤
.
E (aξ + bη)
2m
Taking the limit m → ∞ completes the proof.
3.1.5 Basic properties of expectations
3
Suppose that ξ and η are independent, and
expectations Eξ and Eη exists. Then
Eξη = EξEη.
3.1.5 Basic properties of expectations
3
Suppose that ξ and η are independent, and
expectations Eξ and Eη exists. Then
Eξη = EξEη.
3.1.5 Basic properties of expectations
3
Suppose that ξ and η are independent, and
expectations Eξ and Eη exists. Then
Eξη = EξEη.
Proof. Suppose first ξ ≥ 0 and η ≥ 0. Write
xi =
i
.
2m
The possible values of ξ (m) η (m) are those xi xj s.
3.1.5 Basic properties of expectations
E(ξ (m) η (m) ) =
X
zl P (ξ (m) η (m) = zl )
l
=
=
=
=
X
X
zl
P (ξ (m) = xi , η (m) = xj )
i,j:xi xj =zl
l
∞
∞
XX
xi xj P (ξ (m) = xi , η (m) = xj )
i=0 j=0
∞ X
∞
X
xi xj P (ξ (m) = xi )P (η (m) = xj )
i=0 j=0
∞
X
xi P (ξ
(m)
i=0
=Eξ
(m)
= xi )
∞
X
j=0
Eη
(m)
.
xj P (η (m) = xj )
3.1.5 Basic properties of expectations
By the definition of the expectation of positive r.v.,
Eξ (m) Eη (m) % EξEη.
3.1.5 Basic properties of expectations
By the definition of the expectation of positive r.v.,
Eξ (m) Eη (m) % EξEη.
On the other hand,
ξ (m) η (m) % ξη, (ξη)(m) % ξη.
By Property 5 for discrete random variable and the
definition of the expectation of positive r.v.,
lim E(ξ (m) η (m) ) = lim E (ξη)(m) = E(ξη).
m
m
3.1.5 Basic properties of expectations
By the definition of the expectation of positive r.v.,
Eξ (m) Eη (m) % EξEη.
On the other hand,
ξ (m) η (m) % ξη, (ξη)(m) % ξη.
By Property 5 for discrete random variable and the
definition of the expectation of positive r.v.,
lim E(ξ (m) η (m) ) = lim E (ξη)(m) = E(ξη).
m
m
It follows that
E(ξη) = EξEη.
3.1.5 Basic properties of expectations
For general random variables ξ and η,
ξη = ξ + η + − ξ + η − − ξ − η + + ξ − η − .
3.1.5 Basic properties of expectations
For general random variables ξ and η,
ξη = ξ + η + − ξ + η − − ξ − η + + ξ − η − .
So
E(ξη) =E(ξ + η + ) − E(ξ + η − ) − E(ξ − η + ) + E(ξ − η − )
=Eξ + Eη + − Eξ + Eη − − Eξ − Eη + + Eξ − Eη −
=E(ξ + − ξ − )E(η + − η − ) = EξEη.