Journal of Inequalities in Pure and
Applied Mathematics
SOME INEQUALITIES FOR THE EXPECTATION AND VARIANCE OF
A RANDOM VARIABLE WHOSE PDF IS n-TIME DIFFERENTIABLE
volume 1, issue 2, article 21,
2000.
NEIL S. BARNETT, PIETRO CERONE, SEVER S. DRAGOMIR AND
JOHN ROUMELIOTIS
School of Communications and Informatics
Victoria University of Technology
PO Box 14428, Melbourne City MC
8001 Victoria, Australia
Received 25 February, 2000;
accepted 29 May, 2000.
Communicated by: S.P. Singh
EMail: Neil.Barnett@vu.edu.au
URL: http://sci.vu.edu.au/staff/neilb.html
Abstract
EMail: Peter.Cerone@vu.edu.au
URL: http://sci.vu.edu.au/staff/peterc.html
Contents
EMail: Sever.Dragomir@vu.edu.au
URL: http://rgmia.vu.edu.au/SSDragomirWeb.html
EMail: John.Roumeliotis@vu.edu.au
URL: http://www.staff.vu.edu.au/johnr/
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2000
School of Communications and Informatics,Victoria University of Technology
ISSN (electronic): 1443-5756
005-00
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Abstract
Some inequalities for the expectation and variance of a random variable whose
p.d.f. is n-time differentiable are given.
2000 Mathematics Subject Classification: 60E15, 26D15
Key words: Random Variable, Expectation, Variance, Dispertion
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Some Preliminary Integral Identities . . . . . . . . . . . . . . . . . . . . . 6
3
Some Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
References
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis
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Page 2 of 29
J. Inequal. Pure and Appl. Math. 1(2) Art. 21, 2000
1.
Introduction
Let f : [a, b] → R+ be the p.d.f. of the random variable X and
Z
b
E (X) :=
tf (t) dt
a
its expectation and
Z
σ (X) =
b
21
12 Z b
(t − E (X))2 f (t) dt =
t2 f (t) dt − [E (X)]2
a
a
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis
its dispersion or standard deviation.
In [1], using the identity
(1.1)
2
2
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
Z
[x − E (X)] + σ (X) =
b
(x − t)2 f (t) dt
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a
Contents
and applying a variety of inequalities such as: Hölder’s inequality, pre-Grüss,
pre-Chebychev, pre-Lupaş, or Ostrowski type inequalities, a number of results
concerning the expectation and variance of the random variable X were obtained.
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J. Inequal. Pure and Appl. Math. 1(2) Art. 21, 2000
For example,
(1.2) σ 2 (X) + [x − E (X)]2

h
i
(b−a)2
a+b 2

kf k∞ , if f ∈ L∞ [a, b] ;
(b − a) 12 + x − 2






 h
i1
(b−x)2q+1 +(x−a)2q+1 q
≤
kf kp ,
if f ∈ Lp [a, b] ,

2q+1



p > 1, p1 + 1q = 1;



 b−a 2,
+ x − a+b
2
2
for all x ∈ [a, b], which imply, amongst other things, that
(1.3)
0 ≤ σ (X)

h
i1
1
1
(b−a)2

a+b 2 2
2
2

kf
k
+
E
(X)
−
(b
−
a)

∞ , if f ∈ L∞ [a, b] ;
12
2





n
o1
1
≤
[b−E(X)]2q+1 +[E(X)−a]2q+1 2q
2
kf
k
if f ∈ Lp [a, b] ,

p ,
2q+1




p > 1, p1 + 1q = 1;


 b−a ,
+ E (X) − a+b
2
2
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis
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and
1
(b − a)2 .
4
In this paper more accurate inequalities are obtained by assuming that the
p.d.f. of X is n-time differentiable and that f (n) is absolutely continuous on
(1.4)
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
0 ≤ σ 2 (X) ≤ [b − E (X)] [E (X) − a] ≤
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J. Inequal. Pure and Appl. Math. 1(2) Art. 21, 2000
[a, b]. For other recent results on the application of Ostrowski type inequalities
in Probability Theory, see [2]-[4].
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis
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J. Inequal. Pure and Appl. Math. 1(2) Art. 21, 2000
2.
Some Preliminary Integral Identities
The following lemma, which is interesting in itself, holds.
Lemma 2.1. Let X be a random variable whose probability distribution function f : [a, b] → R+ is n-time differentiable and f (n) is absolutely continuous
on [a, b]. Then
2
2
(2.1) σ (X) + [E (X) − x] =
n
X
(b − x)k+3 + (−1)k (x − a)k+3
f (k) (x)
(k + 3) k!
k=0
Z t
Z b
1
2
n (n+1)
+
(t − x)
(t − s) f
(s) ds dt
n! a
x
for all x ∈ [a, b].
Proof. Is by Taylor’s formula with integral remainder. Recall that
(2.2)
f (t) =
n
X
(t − x)k
k=0
k!
f
(k)
t
Z
1
(x) +
n!
(t − s)n f (n+1) (s) ds
x
for all t, x ∈ [a, b].
Together with
(2.3)
2
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis
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2
Z
σ (X) + [E (X) − x] =
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b
2
(t − x) f (t) dt,
a
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Page 6 of 29
J. Inequal. Pure and Appl. Math. 1(2) Art. 21, 2000
where f is the p.d.f. of the random variable X, we obtain
(2.4)
σ 2 (X) + [E (X) − x]2
" n
#
Z b
Z t
k
X
(t
−
x)
1
=
(t − x)2
f (k) (x) +
(t − s)n f (n+1) (s) ds dt
k!
n!
a
x
k=0
Z
n
k+2
b
X
(t − x)
f (k) (x)
=
dt
k!
a
k=0
Z t
Z
1 b
2
n (n+1)
+
(t − x)
(t − s) f
(s) ds dt
n! a
x
and since
Z
a
b
(t − x)k+2
(b − x)k+3 + (−1)k (x − a)k+3
dt =
,
k!
(k + 3) k!
the identity (2.4) readily produces (2.1)
Corollary 2.2. Under the above assumptions, we have
(2.5)
h
i
k
2 X
n
1
+
(−1)
(b − a)k+3
a+b
a+b
2
(k)
σ (X) + E (X) −
=
f
k+3 (k + 3) k!
2
2
2
k=0
!
2 Z t
Z 1 b
a+b
+
t−
(t − s)n f (n+1) (s) ds dt.
a+b
n! a
2
2
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis
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J. Inequal. Pure and Appl. Math. 1(2) Art. 21, 2000
The proof follows by using (2.4) with x =
a+b
.
2
Corollary 2.3. Under the above assumptions,
(2.6) σ 2 (X) +
1
(E (X) − a)2 + (E (X) − b)2
2
"
#
n
X
(b − a)k+3 f (k) (a) + (−1)k f (k) (b)
=
(k + 3) k!
2
k=0
Z bZ b
1
+
K (t, s) (t − s)n f (n+1) (s) dsdt,
n! a a
where
K (t, s) :=



(t−a)2
2
if a ≤ s ≤ t ≤ b,
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis

 − (t−b)2 if a ≤ t < s ≤ b.
2
Proof. In (2.1), choose x = a and x = b, giving
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(2.7) σ 2 (X) + [E (X) − a]2
Z t
Z
n
X
(b − a)k+3 (k)
1 b
2
n (n+1)
=
f (a) +
(t − a)
(t − s) f
(s) ds dt
(k + 3) k!
n! a
a
k=0
Contents
and
(2.8) σ 2 (X) + [E (X) − b]2
Z t
Z
n
X
(−1)k (b − a)k+3 (k)
1 b
2
n (n+1)
=
f (b)+
(t − b)
(t − s) f
(s) ds dt.
(k
+
3)
k!
n!
a
b
k=0
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J. Inequal. Pure and Appl. Math. 1(2) Art. 21, 2000
Adding these and dividing by 2 gives (2.6).
Taking into account that µ = E (X) ∈ [a, b], then we also obtain the following.
Corollary 2.4. With the above assumptions,
2
(2.9) σ (X) =
n
X
(b − µ)k+3 + (−1)k (µ − a)k+3
k=0
f (k) (µ)
(k + 3) k!
Z t
Z
1 b
2
n (n+1)
(t − µ)
(t − s) f
(s) ds dt.
+
n! a
µ
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis
Proof. The proof follows from (2.1) with x = µ ∈ [a, b].
Lemma 2.5. Let the conditions of Lemma 2.1 relating to f hold. Then the
following identity is valid.
(2.10) σ 2 (X) + [E (X) − x]2
Z
n
X
(b − x)k+3 + (−1)k (x − a)k+3 f (k) (x) 1 b
=
·
+
Kn (x, s) f (n+1) (s) ds,
k
+
3
k!
n!
a
k=0
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where
(2.11)
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
K (x, s) =

n+1
ψn (s − a, x − s) , a ≤ s ≤ x
 (−1)

ψn (b − s, s − x) ,
x<s≤b
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J. Inequal. Pure and Appl. Math. 1(2) Art. 21, 2000
with
(2.12) ψn (u, v) =
un+1
· (n + 2) (n + 1) u2
(n + 3) (n + 2) (n + 1)
+2 (n + 3) (n + 1) uv + (n + 3) (n + 2) v 2 .
Proof. From (2.1), an interchange of the order of integration gives
Z
Z t
1 b
2
(t − x) dt
(t − s)n f (n+1) (s) ds
n! a
x
Z xZ s
1
=
−
(t − x)2 (t − s)n f (n+1) (s) dtds
n!
a
a
Z bZ b
2
n (n+1)
+
(t − x) (t − s) f
(s) dtds
x
s
Z
1 b
K̃n (x, s) f (n+1) (s) ds,
=
n! a
where
K̃n (x, s) =
Rs

2
n
 pn (x, s) = − a (t − x) (t − s) dt, a ≤ s ≤ x

qn (x, s) =
Rb
s
(t − x)2 (t − s)n dt,
To prove the lemma it is sufficient to show that K ≡ K̃.
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis
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x < s < b.
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Page 10 of 29
J. Inequal. Pure and Appl. Math. 1(2) Art. 21, 2000
Now,
Z
s
2
n+1
Z
s−a
(t − x) (t − s) dt = (−1)
(u + x − s)2 un du
a
0
Z s−a
= (−1)n+1
u2 + 2 (x − s) u + (x − s)2 un du
p̃n (x, s) = −
n
0
n+1
= (−1)
ψn (s − a, x − s) ,
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
where ψ (·, ·) is as given by (2.12). Further,
Z
b
2
n
Z
(t − x) (t − s) dt =
q̃n (x, s) =
s
b−s
[u + (s − x)]2 un du = ψn (b − s, s − x) ,
0
where, again, ψ (·, ·) is as given by (2.12). Hence K ≡ K̃ and the lemma is
proved.
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis
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J. Inequal. Pure and Appl. Math. 1(2) Art. 21, 2000
3.
Some Inequalities
We are now able to obtain the following inequalities.
Theorem 3.1. Let X be a random variable whose probability density function
f : [a, b] → R+ is n-time differentiable and f (n) is absolutely continuous on
[a, b], then
n
X
(b − x)k+3 + (−1)k (x − a)k+3 (k)
2
2
(3.1) σ (X) + [E (X) − x] −
f (x)
(k + 3) k!
k=0

kf (n+1) k∞ n+4
n+4 

(x
−
a)
+
(b
−
x)
, if f (n+1) ∈ L∞ [a, b] ;

(n+1)!(n+4)






1

n+3+ 1
q +(b−x)n+3+ q
(x−a)
f (n+1) k
k
p
≤
,
if f (n+1) ∈ Lp [a, b] ,
1
1

n!
n+3+
q
(
)
(nq+1)
q




p > 1, p1 + 1q = 1;



(n+1)

k1  kf
(x − a)n+3 + (b − x)n+3 ,
n!(n+3)
for all x ∈ [a, b], where k·kp (1 ≤ p ≤ ∞) are the usual Lebesque norms on
[a, b], i.e.,
Z
kgk∞ := ess sup |g (t)|
t∈[a,b]
and
kgkp :=
a
b
p1
p
|g (t)| dt , p ≥ 1.
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis
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J. Inequal. Pure and Appl. Math. 1(2) Art. 21, 2000
Proof. By Lemma 2.1,
σ 2 (X) + [E (X) − x]2 −
n
X
(b − x)k+3 + (−1)k (x − a)k+3
k=0
b
(3.2)
Z
k! (k + 3)
Z t
2
n (n+1)
(t − x)
(t − s) f
(s) ds dt
1
n! a
:= M (a, b; x) .
=
f (k) (x)
x
Clearly,
Z t
n
(n+1)
(t − x) (t − s) f
(s) ds dt
a
"x
Z t
#
Z b
1
(t − x)2 sup f (n+1) (s) |t − s|n ds dt
n! a
s∈[x,t]
x
(n+1) Z
b
f
(t − x)2 |t − x|n+1
∞
dt
n!
n+1
a
(n+1) Z
b
f
∞
|t − x|n+3 dt
(n + 1)! a
(n+1) Z
Z b
x
f
n+3
n+3
∞
(x − t)
dt +
(t − x)
dt
(n + 1)!
a
x
(n+1) f
(x − a)n+4 + (b − x)n+4
1
|M (a, b; x)| ≤
n!
≤
≤
=
=
=
Z
b
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
2
∞
(n + 1)! (n + 4)
and the first inequality in (3.1) is obtained.
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis
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J. Inequal. Pure and Appl. Math. 1(2) Art. 21, 2000
For the second, we use Hölder’s integral inequality to obtain
1
Z t
1q Z t
(n+1) p p
nq
(s) ds dt
(t − x) |t − s| ds f
a
x
x
p1 Z b
Z b
(n+1) p
nq+1
1
f
≤
(s) ds
(t − x)2 |t − x| q dt
n!
a
a
(n+1)
Z
b
1
1 f
p
=
|t − x|n+2+ q dt
1
n! (nq + 1) q a
(n+1) "
#
(b − x)n+3+ 1q + (x − a)n+3+ 1q
1 f
p
.
=
n! (nq + 1) 1q
n + 3 + 1q
1
|M (a, b; x)| ≤
n!
Z
b
2
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis
Finally, note that
Z t
Z
(n+1) 1 b
2
n
|M (a, b; x)| ≤
(t − x) |t − x| f
(s) ds dt
n! a
x
(n+1) Z
b
f
1
≤
|t − x|n+2 dt
n!
#
(n+1) "a
f
(x − a)n+3 + (b − x)n+3
1
=
n!
n+3
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and the third part of (3.1) is obtained.
It is obvious that the best inequality in (3.1) is when x =
lary 3.2.
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a+b
,
2
giving Corol-
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J. Inequal. Pure and Appl. Math. 1(2) Art. 21, 2000
Corollary 3.2. With the above assumptions on X and f ,
(3.3)
h
i
k
k+3
2 X
n
1
+
(−1)
(b
−
a)
2
a + b σ (X) + E (X) − a + b −
f (k)
k+3
2
2 (k + 3) k!
2
k=0

kf (n+1) k∞

(b − a)n+4 , if f (n+1) ∈ L∞ [a, b] ;

n+3

2
(n+1)!(n+4)






n+3+ 1
kf (n+1) kp
q
(b−a)
, if f (n+1) ∈ Lp [a, b] , p > 1,
≤
1
n+2+ 1
q
q n! n+3+ 1
(nq+1)
2

(
q)


1

+ 1q = 1;

p



 kf (n+1) k1
(b − a)n+3 .
2n+2 n!(n+3)
The following corollary is interesting as it provides the opportunity to approximate the variance when the values of f (k) (µ) are known, k = 0, ..., n.
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis
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J. Inequal. Pure and Appl. Math. 1(2) Art. 21, 2000
Corollary 3.3. With the above assumptions and µ =
a+b
,
2
we have
n
k+3
k
k+3
X
+
(−1)
(µ
−
a)
(b
−
µ)
(3.4) σ 2 (X) −
f (k) (µ)
(k
+
3)
k!
k=0

kf (n+1) k∞ n+4
n+4 

(µ
−
a)
+
(b
−
µ)
, if f (n+1) ∈ L∞ [a, b] ;

(n+1)!(n+4)


1

n+3+ 1
q +(b−µ)n+3+ q

 kf (n+1) kp (µ−a)
,
if f (n+1) ∈ Lp [a, b] ,
1
≤
n!(n+3+ 1q )
q
(nq+1)



p > 1, p1 + 1q = 1;



(n+1)

k1  kf
(µ − a)n+3 + (b − µ)n+3 .
n!(n+3)
The following result also holds.
Theorem 3.4. Let X be a random variable whose probability density function
f : [a, b] → R+ is n-time differentiable and f (n) is absolutely continuous on
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis
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J. Inequal. Pure and Appl. Math. 1(2) Art. 21, 2000
[a, b], then
1
(3.5) σ 2 (X) + (E (X) − a)2 + (E (X) − b)2
2
"
#
n
X
(b − a)k+3 f (k) (a) + (−1)k f (k) (b) −
(k
+
3)
k!
2
k=0

1
f (n+1) (b − a)n+4 ,
if f (n+1) ∈ L∞ [a, b] ;

(n+4)(n+1)!
∞


1
(n+1) (b−a)n+3+ q


21/q−1
f
, if f (n+1) ∈ Lp [a, b] ,
1
1
1
p
q
q
q
≤
(nq+1)
n!(qn+1) [(n+2)q+2]


p > 1, p1 + 1q = 1;


 1 (n+1) f
(b − a)n+3 ,
2n!
1
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis
where k·kp (1 ≤ p ≤ ∞) are the usual Lebesque p−norms.
Title Page
Proof. Using Corollary 2.3
2
σ (X) + 1 (E (X) − a)2 + (E (X) − b)2
2
"
#
n
X
(b − a)k+3 f (k) (a) + (−1)k f (k) (b) −
(k
+
3)
k!
2
k=0
Z Z
1 b b
≤
|K (t, s)| |t − s|n f (n+1) (s) dsdt
n! a a
=: N (a, b) .
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J. Inequal. Pure and Appl. Math. 1(2) Art. 21, 2000
It is obvious that
N (a, b)
Z Z
(n+1) 1 b b
≤ f
|K (t, s)| |t − s|n dsdt
∞ n!
a
a
Z Z t
Z b
(n+1) 1 b
n
n
|K (t, s)| |t − s| ds +
|K (t, s)| |t − s| ds dt
= f
∞ n!
a
a
t
#
Z b"
2
n+1
2
n+1
1 (t
−
a)
(t
−
a)
(t
−
b)
(b
−
t)
f (n+1) dt
·
+
·
=
∞
n!
2
n+1
2
n+1
a
Z b
(n+1) 1
=
f
(t − a)n+3 + (b − t)n+3 dt
∞
2 (n + 1)!
"a
#
n+4
n+4
(n+1) (b
−
a)
(b
−
a)
1
f
+
=
∞
2 (n + 1)!
n+4
n+4
(n+1) f
∞
=
(b − a)n+4
(n + 4) (n + 1)!
so the first part of (3.5) is proved.
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis
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J. Inequal. Pure and Appl. Math. 1(2) Art. 21, 2000
Using Hölder’s integral inequality for double integrals,
N (a, b)
Z b Z b
p1 Z b Z b
1q
(n+1) p
1
q
qn
f
≤
(s) dsdt
×
|K (t, s)| |t − s| dsdt
n!
a
a
a
a
1 (b − a) p f (n+1) p Z b Z t
=
|K (t, s)|q |t − s|qn ds
n!
a
a
1q
Z b
q
qn
+
|K (t, s)| |t − s| ds dt
t
1
p
"
"
(b − a) f (n+1) p Z b (t − a)2q Z t
|t − s|qn ds
=
q
n!
2
a
a
# # 1q
Z
(t − b)2q b
+
|t − s|qn ds dt
2q
t
"
"
# # 1q
1 (b − a) p f (n+1) p Z b (t − a)2q (t − a)qn+1 (t − b)2q (b − t)qn+1
=
+
dt
n!
2q (qn + 1)
2q (qn + 1)
a
1 1q Z b
(b − a) p f (n+1) p 1
=
· q
(t − a)(n+2)q+1 dt
n!
2 (qn + 1)
a
1q
Z b
+
(b − t)(n+2)q+1 dt
a
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis
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J. Inequal. Pure and Appl. Math. 1(2) Art. 21, 2000
# 1q
1 1q "
(b − a) p f (n+1) p (b − a)(n+2)q+2 (b − a)(n+2)q+2
1
=
· q
+
n!
2 (qn + 1)
(n + 2) q + 2
(n + 2) q + 2
1
2
21/q f (n+1) p (b − a)n+2+ p + q
=
1
1
n!2 (qn + 1) q ((n + 2) q + 2) q
i
h
n+3+ 1q
1/q−1 (n+1) 2
f
(b − a)
p
=
1
1
n! (qn + 1) q [(n + 2) q + 2] q
and the second part of (3.5) is proved.
Finally, we observe that
N (a, b) ≤
1
sup |K (t, s)| |t − s|n
n! (t,s)∈[a,b]2
2
1 (b − a)
· (b − a)n (b − a)
n!
2
1
=
(b − a)n+3 f (n+1) 1 ,
2n!
Z bZ
a
Z
=
b
(n+1) f
(s) dsdt
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis
a
Title Page
b
(n+1) f
(s) ds
a
which is the final result of (3.5).
The following particular case can be useful in practical applications. For
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J. Inequal. Pure and Appl. Math. 1(2) Art. 21, 2000
n = 0, (3.1) becomes
(3.6)
#
"
2
a+b
(b − a)2
2
2
f (x)
+
σ (X) + [E (X) − x] − (b − a) x −
2
12
 kf 0 k ∞
(x − a)4 + (b − x)4 ,
if f 0 ∈ L∞ [a, b] ;


4




i
 qkf 0 kp h
3+ 1q
3+ 1q
(x − a)
+ (b − x)
, if f 0 ∈ Lp [a, b] ,
≤
3q+1


p > 1, p1 + 1q = 1;


h
i


 kf 0 k (b−a)2 + x − a+b 2 ,
1
12
for all x ∈ [a, b]. In particular, for x =
(3.7)
2
a+b
,
2
2
a+b
(b − a)3
a + b 2
−
f
σ (X) + E (X) −
2
12
2
 0
kf k∞

(b − a)4 , if f 0 ∈ L∞ [a, b] ;

32






3+ 1
qkf 0 kp (b−a) q
≤
, if f 0 ∈ Lp [a, b] ,
2+ 1
q (3q+1)

2



p > 1, p1 + 1q = 1;


 kf 0 k1

(b − a)3 ,
12
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis
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which is, in a sense, the best inequality that can be obtained from (3.6). If in
Page 21 of 29
J. Inequal. Pure and Appl. Math. 1(2) Art. 21, 2000
(3.6) x = µ = E (X), then
"
#
2
a+b
(b − a)2
2
(3.8) σ (X) − (b − a) E (X) −
+
f (E (X))
2
12
 kf 0 k ∞
(E (X) − a)4 + (b − E (X))4 , if f 0 ∈ L∞ [a, b] ;

4





 kf 0 kp 4
4
(E
(X)
−
a)
+
(b
−
E
(X))
, if f 0 ∈ Lp [a, b] , p > 1,
≤
3+ 1q )
(




h
i


 kf 0 k (b−a)2 + E (X) − a+b 2 .
1
12
2
In addition, from (3.5),
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis
(3.9)
3 1
(b
−
a)
f
(a)
+
f
(b)
2
2
2
σ (X) + (E (X) − a) + (E (X) − b) −
2
3
2
 1 0
kf k∞ (b − a)4 ,
if f 0 ∈ L∞ [a, b] ;

4





3+ 1q
1
0
, if f 0 ∈ Lp [a, b] , p > 1,
1
1 kf kp (b − a)
≤
n!2 q (q+1) q





 1 0
kf k1 (b − a)3 ,
2
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which provides an approximation for the variance in terms of the expectation
and the values of f at the end points a and b.
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J. Inequal. Pure and Appl. Math. 1(2) Art. 21, 2000
Theorem 3.5. Let X be a random variable whose p.d.f. f : [a, b] → R+ is
n−time differentiable and f (n) is absolutely continuous on [a, b]. Then
(3.10)
n
k+3
k
k+3
(k)
X
(b − x)
+ (−1) (x − a)
f (x) 2
2
·
σ (X) + (E (X) − x) −
k+3
k! k=0
 kf (n+1) k

∞

(x − a)n+4 + (b − x)n+4 (n+1)!(n+4)
,
if f (n+1) ∈ L∞ [a, b] ;






 1h
i 1q kf (n+1) k
f (n+1) ∈Lp [a, b] ,
p
,
if
≤
C q (x − a)(n+3)q+1 + (b − x)(n+3)q+1
n!
p > 1;







n+3 kf (n+1) k

 b−a 1
+ x − a+b ·
,
2
2
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis
n!(n+3)
where
Title Page
Z
(3.11)
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
C=
0
1
n+3
n+2
n+1
u
u
u
+ 2 (1 − u)
+ (1 − u)2
n+3
n+2
n+1
q
du.
Proof. From (2.10),
(3.12)
n
X
(b − x)k+3 + (−1)k (x − a)k+3 f (k) (x) 2
2
·
σ (X) + (E (X) − x) −
k+3
k! k=0
Z b
1
(n+1)
= Kn (x, s) f
(s) ds .
n! a
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J. Inequal. Pure and Appl. Math. 1(2) Art. 21, 2000
Now, on using the fact that from (2.11), (2.12), ψn (u, v) ≥ 0 for u, v ≥ 0,
Z b
1
(n+1)
(3.13) Kn (x, s) f
(s) ds
n! a
(n+1) Z
Z b
x
f
∞
≤
ψn (s − a, x − s) ds +
ψn (b − s, s − x) ds .
n!
a
x
Further,
(3.14)
un+2
un+1
un+3
ψn (u, v) =
+ 2v
+ v2
n+3
n+2
n+1
and so
(3.15)
Z x
ψn (s − a, x − s) ds
a
#
Z x"
n+1
(s − a)n+3
(s − a)n+2
2 (s − a)
+ 2 (x − s)
+ (x − s)
ds
=
n+3
n+2
n+1
a
Z 1 n+3
n+1
λ
λn+2
n+4
2 λ
= (x − a)
+ 2 (1 − λ)
+ (1 − λ)
dλ,
n+3
n+2
n+1
0
s−a
.
x−a
where we have made the substitution λ =
Collecting powers of λ gives
1
2
1
2λn+2
λn+1
n+3
λ
−
+
−
+
n+3 n+2 n+1
(n + 2) (n + 1) n + 1
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis
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J. Inequal. Pure and Appl. Math. 1(2) Art. 21, 2000
and so, from (3.15),
Z x
(3.16)
ψn (s − a, x − s) ds
a
1
1
2
1
= (x − a)
−
+
n+4 n+3 n+2 n+1
2
1
−
+
(n + 3) (n + 2) (n + 1) (n + 2) (n + 1)
n+4
(x − a)n+4
.
=
(n + 4) (n + 1)
Similarly, on using (3.14),
Z
x
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis
b
ψn (b − s, s − x) ds
#
Z b"
n+1
(b − s)n+2
(b − s)n+3
(b
−
s)
+ 2 (s − x)
+ (s − x)2
ds
=
n
+
3
n
+
2
n+1
x
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J. Inequal. Pure and Appl. Math. 1(2) Art. 21, 2000
and making the substitution ν =
b−s
b−x
gives
(3.17)
Z b
ψn (b − s, s − x) ds
x
n+4
Z
= (b − x)
0
n+4
=
1
n+1
ν n+2
ν n+3
2 ν
+ 2 (1 − ν)
+ (1 − ν)
dν
n+3
n+2
n+1
(b − x)
,
(n + 4) (n + 1)
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
where we have used (3.15) and (3.16). Combining (3.16) and (3.17) gives the
first inequality in (3.10). For the second inequality in (3.10), we use Hölder’s
integral inequality to obtain
(3.18)
Z b
1q
f (n+1) (s) Z b
1
p
q
(n+1)
Kn (x, s) f
(s) ds ≤
|Kn (x, s)| ds .
n!
n!
a
a
Now, from (2.11) and (3.14)
Z b
Z x
Z b
q
q
|Kn (x, s)| ds =
ψ (s − a, x − s) ds +
ψ q (b − s, s − x) ds
a
a
x
h
i
(n+3)q+1
= C (x − a)
+ (b − x)(n+3)q+1 ,
where C is as defined in (3.11) and we have used (3.15) and (3.16). Substitution
into (3.18) gives the second inequality in (3.10).
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis
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J. Inequal. Pure and Appl. Math. 1(2) Art. 21, 2000
Finally, for the third inequality in (3.10). From (3.12),
(3.19)
Z b
1
(n+1)
K
(x,
s)
f
(s)
ds
n
n!
a
Z x
Z b
(n+1) (n+1) 1
≤
ψn (s − a, x − s) f
(s) ds +
ψn (b − s, s − x) f
(s) ds
n!
a
x
Z x
Z b
(n+1) (n+1) 1
f
f
≤
ψn (x − a, 0)
(s) ds + ψn (b − x, 0)
(s) ds ,
n!
a
x
where, from (3.14),
(3.20)
ψn (u, 0) =
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis
un+3
.
n+3
Hence, from (3.19) and (3.20)
)
(
Z b
n+3
n+3
1
(n+1) 1
(x
−
a)
(b
−
x)
f
Kn (x, s) f (n+1) (s) ds ≤
max
,
(·)1
n!
n!
n+3
n+3
a
1
=
[max {x − a, b − x}]n+3 f (n+1) (·)1 ,
n! (n + 3)
which, on using the fact that for X, Y ∈ R
X − Y
X +Y
max {X, Y } =
+ 2
2
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
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gives, from (3.12), the third inequality in (3.10). The theorem is now completely
proved.
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J. Inequal. Pure and Appl. Math. 1(2) Art. 21, 2000
Remark 3.1. The results of Theorem 3.5 may be compared with those of Theorem 3.1. Theorem 3.5 is based on the single integral identity developed in
Lemma 2.5, while Theorem 3.1 is based on the double integral identity representation for the bound. It may be noticed from (3.1) and (3.10) that the bounds
are the same for f (n+1) ∈ L∞ [a, b], that for f (n+1) ∈ L1 [a, b] (which is always
true since f (n) is absolutely continuous) the bound obtained in (3.1) is better
and for f (n+1) ∈ Lp [a, b], p > 1, the result is inconclusive.
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis
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References
[1] N.S. BARNETT, P. CERONE, S.S. DRAGOMIR AND J. ROUMELIOTIS,
Some inequalities for the dispersion of a random variable whose p.d.f. is
defined on a finite interval, J. Inequal. Pure and Appl. Math., accepted for
publication.
[2] N.S. BARNETT AND S.S. DRAGOMIR, An inequality of Ostrowski type
for cumulative distribution functions, Kyungpook Math. J., 39(2) (1999),
303–311.
[ONLINE] (Preprint available) http://rgmia.vu.edu.au/v1n1.html
[3] N.S. BARNETT AND S.S. DRAGOMIR, An Ostrowski type inequality for
a random variable whose probability density function belongs to L∞ [a, b],
Nonlinear Anal. Forum, 5 (2000), 125–135.
[ONLINE] (Preprint available) http://rgmia.vu.edu.au/v1n1.html
Some Inequalities for the
Expectation and Variance of a
Random Variable whose PDF is
n-Time Differentiable
Neil S. Barnett, Pietro Cerone,
Sever S. Dragomir and
John Roumeliotis
Title Page
Contents
[4] S.S. DRAGOMIR, N.S. BARNETT AND S. WANG, An Ostrowski type
inequality for a random variable whose probability density function belongs
to Lp [a, b], p > 1, Math. Inequal. Appl., 2(4) (1999), 501–508.
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