Journal of Inequalities in Pure and
Applied Mathematics
MOMENTS INEQUALITIES OF A RANDOM VARIABLE DEFINED
OVER A FINITE INTERVAL
volume 3, issue 3, article 41,
2002.
PRANESH KUMAR
Department of Mathematics & Computer Science
University of Northern British Columbia
Prince George
BC V2N 4Z9, Canada.
EMail: kumarp@unbc.ca
Received 01 October, 2001;
accepted 03 April, 2002.
Communicated by: C.E.M. Pearce
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
Some estimations and inequalities are given for the higher order central moments of a random variable taking values on a finite interval. An application is
considered for estimating the moments of a truncated exponential distribution.
2000 Mathematics Subject Classification: 60 E15, 26D15.
Key words: Random variable, Finite interval, Central moments, Hölder’s inequality,
Grüss inequality.
Moments Inequalities of A
Random Variable Defined Over
A Finite Interval
This research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. Thanks are due to the referee and Prof. Sever
Dragomir for their valuable comments that helped in improving the paper.
Pranesh Kumar
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Contents
Contents
1
2
3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Results Involving Higher Moments . . . . . . . . . . . . . . . . . . . . . .
Some Estimations for the Central Moments . . . . . . . . . . . . . . .
3.1
Bounds for the Second Central Moment M2 (Variance) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Bounds for the Third Central Moment M3 . . . . . . . . .
3.3
Bounds for the Fourth Central Moment M4 . . . . . . . .
4
Results Based on the Grüss Type Inequality . . . . . . . . . . . . . .
5
Results Based on the Hölder’s Integral Inequality . . . . . . . . . .
6
Application to the Truncated Exponential Distribution . . . . . .
References
3
4
6
8
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1.
Introduction
Distribution functions and density functions provide complete descriptions of
the distribution of probability for a given random variable. However they do
not allow us to easily make comparisons between two different distributions.
The set of moments that uniquely characterizes the distribution under reasonable conditions are useful in making comparisons. Knowing the probability
function, we can determine the moments, if they exist. There are, however, applications wherein the exact forms of probability distributions are not known or
are mathematically intractable so that the moments can not be calculated. As
an example, an application in insurance in connection with the insurer’s payout on a given contract or group of contracts follows a mixture or compound
probability distribution that may not be known explicitly. It is this problem that
motivates to find alternative estimations for the moments of a probability distribution. Based on the mathematical inequalities, we develop some estimations
of the moments of a random variable taking its values on a finite interval.
Set X to denote a random variable whose probability function is f : [a, b] ⊂
R → R+ and its associated distribution function F : [a, b] → [0, 1].
Denote by Mr the rth central moment of the random variable X defined as
Z b
(1.1)
Mr =
(t − µ)r dF, r = 0, 1, 2, . . . ,
a
where µ is the mean of the random variable X. It may be noted that M0 = 1,
M1 = 0 and M2 = σ 2 , the variance of the random variable X.
When reference is made to the rth moment of a particular distribution, we
assume that the appropriate integral (1.1) converges for that distribution.
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2.
Results Involving Higher Moments
We first prove the following theorem for the higher central moments of the
random variable X.
Theorem 2.1. For the random variable X with distribution function F : [a, b] →
[0, 1],
Z b
(2.1)
(b − t)(t − a)m dF
a
m X
m
=
(µ − a)k [(b − µ)Mm−k − Mm−k+1 ] , m = 1, 2, 3, . . . .
k
k=0
Proof. Expressing the left hand side of (2.1) as
Z b
Z b
m
(b − t)(t − a) dF =
[(b − µ) − (t − µ)][(t − µ) + (µ − a)]m dF,
a
a
and using the binomial expansion
[(t − µ) + (µ − a)]m =
m X
k=0
m
(µ − a)k (t − µ)m−k ,
k
a
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[(b − µ) − (t − µ)]
=
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b
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we get
Z b
(b − t)(t − a)m dF
Z
Moments Inequalities of A
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m X
m
k=0
k
#
(µ − a)k (t − µ)m−k dF
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=
−
m X
m
k=0
m X
k=0
k
k
Z
(b − µ)(µ − a) ·
b
(t − µ)m−k dF
a
m
(µ − a)k ·
k
Z
b
(t − µ)m−k+1 dF,
a
and hence the theorem.
In practice numerical moments of order higher than the fourth are rarely considered, therefore, we now derive the results for the first four central moments
of the random variable X based on Theorem 2.1.
Corollary 2.2. For m = 1, k = 0, 1 in (2.1), we have
Z b
(2.2)
(b − t)(t − a)dF = (b − µ)(µ − a) − M2 .
a
This is a result in Theorem 1 by Barnett and Dragomir [1].
Corollary 2.3. For m = 2, k = 0, 1, 2 in (2.1),
Z b
(2.3)
(b − t)(t − a)2 dF = (b − µ)(µ − a)2 + [(b − µ) − 2(µ − a)]M2 − M3 .
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Corollary 2.4. For m = 3, k = 0, 1, 2, 3, we have from (2.1)
Z b
(2.4)
(b − t)(t − a)3 dF
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a
= (b − µ)(µ − a)3 + 3(µ − a)[(b − µ) − (µ − a)]M2
+ [(b − µ) − 3(µ − a)]M3 − M4 .
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3.
Some Estimations for the Central Moments
We apply Hölder’s inequality [4] and results of Barnett and Dragomir [1] to
derive the bounds for the central moments of the random variable X.
Theorem 3.1. For the random variable X with distribution function F : [a, b] →
[0, 1], we have


r+s+1 Γ(r + 1)Γ(s + 1)

Z b
·
· ||f ||∞ ,
 (b − a)
Γ(r + s + 2)
r
s
(3.1)
(b − t) (t − a) dF ≤

a

1

(b − a)2+ q [B(rq + 1, sq + 1)] · ||f ||p ,
Moments Inequalities of A
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Pranesh Kumar
for p > 1,
1
p
+
1
q
= 1, r, s ≥ 0.
Proof. Let t = a(1 − u) + bu. Then
Z b
Z 1
r
s
r+s+1
(b − t) (t − a) dt = (b − a)
·
(1 − u)r us du.
a
Since
R1
0
0
us (1 − u)r du =
Z
Γ(r+1)Γ(s+1)
,
Γ(r+s+2)
b
(b − t)r (t − a)s dt = (b − a)r+s+1 ·
a
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Γ(r + 1)Γ(s + 1)
.
Γ(r + s + 2)
Using the property of definite integral,
Z b
(3.2)
(b − t)s (t − a)r dF ≥ 0, for r, s ≥ 0,
a
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we get,
Z
a
b
(b − t)s (t − a)r dF
Z b
≤ ||f ||∞
(b − t)s (t − a)r dt,
a
= (b − a)r+s+1 ·
Γ(r + 1)Γ(s + 1)
· ||f ||∞ for r, s ≥ 0,
Γ(r + s + 2)
the first inequality in (3.1).
Now applying the Hölder’s integral inequality,
Z b
(b − t)s (t − a)r dF
a
Z
≤
b
1q
p1 Z b
(b − t)sq (t − a)rq dt
f p (t)dt
a
1
the second inequality in (3.1).
Theorem 3.2. For the random variable X with distribution function F : [a, b] →
[0, 1],
Γ(r + 1)Γ(s + 1)
m(b − a)r+s+1 ·
Γ(r + s + 2)
Z b
≤
(b − t)s (t − a)r dF
a
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a
= (b − a)2+ q [B(rq + 1, sq + 1)] · ||f ||p ,
(3.3)
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≤ M (b − a)r+s+1 ·
Γ(r + 1)Γ(s + 1)
, r, s ≥ 0.
Γ(r + s + 2)
Proof. Noting that if m ≤ f ≤ M, a.e. on [a, b], then
m(b − t)s (t − a)r ≤ (b − t)s (t − a)r f ≤ M (b − t)s (t − a)r ,
a.e. on [a, b] and by integrating over [a, b], we prove the theorem.
3.1.
Bounds for the Second Central Moment M2 (Variance)
Moments Inequalities of A
Random Variable Defined Over
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It is seen from (2.2) and (3.2) that the upper bound for M2 , variance of the
random variable X, is
Pranesh Kumar
M2 ≤ (b − µ)(µ − a).
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(3.4)
Considering x = (b − µ) and y = (µ − a) in the elementary result
xy ≤
(x + y)2
, x, y ∈ R,
4
we have
(3.5)
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(b − a)2
M2 ≤
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and thus,
(3.6)
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0 ≤ M2 ≤ (b − µ)(µ − a) ≤
(b − a)2
.
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From (2.2) and (3.1), we get
(b − µ)(µ − a) − M2 ≤
(b − a)3
||f ||∞ ,
6
1
(b − µ)(µ − a) − M2 ≤ ||f ||p (b − a)2+ q [B(q + 1, q + 1)], p > 1,
1 1
+ = 1.
p q
Other estimations for M2 from (2.2) and (3.1) are
m
(b − a)3
(b − a)3
≤ (b − µ)(µ − a) − M2 ≤ M
, m ≤ f ≤ M,
6
6
Moments Inequalities of A
Random Variable Defined Over
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resulting in
Pranesh Kumar
(3.7)
3.2.
(b − a)3
M2 ≤ (b − µ)(µ − a) − m
, m ≤ f ≤ M.
6
Bounds for the Third Central Moment M3
From (2.3) and (3.2), the upper bound for M3
M3 ≤ (b − µ)(µ − a)2 + [(b − µ) − 2(µ − a)]M2 .
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Further we obtain from (2.3) and (3.4),
M3 ≤ (b − µ)(µ − a)(a + b − 2µ),
(3.8)
from (2.3) and (3.5),
(3.9)
1
M3 ≤ [(b − µ)3 + (b − µ)(µ − a)2 − 2(µ − a)3 ],
4
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and from (2.3) and (3.7),
(3.10)
3.3.
M3 ≤ (b − µ)(µ − a)(a + b − 2µ) −
m(b − a)3 (b + µ − 2a)
.
6
Bounds for the Fourth Central Moment M4
The upper bounds for M4 from (2.4) and (3.2)
M4 ≤ (b − µ)(µ − a)3
+ 3(µ − a)[(b − µ) − (µ − a)]M2 + [(b − µ) − 3(µ − a)]M3 .
Using (2.4), (3.4) and (3.8), we have
(3.11)
Moments Inequalities of A
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Pranesh Kumar
M4 ≤ (b − µ)(µ − a)[(b − a)2 − 3(b − µ)(µ − a)],
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from (2.4), (3.5) and (3.9),
(3.12) M4 ≤
1
(b − µ)4 + 4(b − µ)2 (µ − a)2
4
−4(b − µ)(µ − a)3 + 3(µ − a)4 ,
and from (2.4), (3.7) and (3.10),
(3.13) M4 ≤ (b − µ)(µ − a)[(µ − a)2
+ (a + b − 2µ)(a + b − 4µ) + 3(b − µ)(a + b − 2µ)]
m(b − a)3 (a + b − 2µ)(b − 2a − µ)
−
.
6
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4.
Results Based on the Grüss Type Inequality
We prove the following theorem based on the pre-Grüss inequality:
Theorem 4.1. For the random variable X with distribution function F : [a, b] →
[0, 1],
(4.1)
Z b
Γ(r
+
1)Γ(s
+
1)
r
s
r+s
(b − t) (t − a) f (t)dt − (b − a) ·
Γ(r + s + 2) a
1
≤ (M − m)(b − a)r+s+1
2
"
2 # 12
Γ(r + 1)Γ(s + 1)
Γ(2r + 1)Γ(2s + 1)
,
×
−
Γ(2r + 2s + 2)
Γ(r + s + 2)
Moments Inequalities of A
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Pranesh Kumar
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where m ≤ f ≤ M a.e. on [a, b] and r, s ≥ 0.
Proof. We apply the following pre-Grüss inequality [4]:
(4.2)
Z b
Z b
Z b
1
1
h(t)g(t)dt −
h(t) dt ·
g(t)dt
b−a a
b−a a
a
"
2 # 12
Z b
Z b
1
1
1
≤ (φ − γ) ·
g 2 (t)dt −
g(t)dt
,
2
b−a a
b−a a
provided the mappings h, g : [a, b] → R are measurable, all integrals involved
exist and are finite and γ ≤ h ≤ φ a.e. on [a, b].
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Let h(t) = f (t), g(t) = (b − t)r (t − a)s in (4.2). Then
(4.3)
Z b
(b − t)r (t − a)s f (t)dt
a
Z b
Z b
1
1
r
s −
f (t)dt ·
(b − t) (t − a) dt
b−a a
b−a a
Z b
1
1
≤ (M − m) ·
{(b − t)r (t − a)s }2 dt
2
b−a a
2 # 12
Z b
1
,
−
(b − t)r (t − a)s dt
b−a a
where m ≤ f ≤ M a.e. on [a.b].
On substituting from (3.2) into (4.3), we prove the theorem.
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Corollary 4.2. For r = s = 1 in (4.2),
Z b
2
(b
−
a)
− a)3
(b − t)(t − a)f (t)dt −
≤ (M − m)(b
√
,
6
12 5
a
a result (2.7) in Theorem 1 by Barnett and Dragomir [1].
We have the following lemma based on the pre-Grüss inequality:
Lemma 4.3. For the random variable X with distribution function F : [a, b] →
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[0, 1],
(4.4)
Z b
(b − t)r (t − a)s f (t)dt − (b − a)r+s Γ(r + 1)Γ(s + 1) Γ(r + s + 2) a
21
Z b
1
2
≤ (M − m) (b − a)
f (t)dt − 1 ,
2
a
where m ≤ f ≤ M a.e. on [a, b] and r, s ≥ 0.
r
s
Proof. We choose h(t) = (b − t) (t − a) , g(t) = f (t) in the pre-Grüss inequality (4.2) to prove this lemma.
Moments Inequalities of A
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We now prove the following theorems based on Lemma 4.3:
Theorem 4.4. For the random variable X with distribution function F : [a, b] →
[0, 1],
(4.5)
Z b
(b − t)r (t − a)s f (t)dt − (b − a)r+s Γ(r + 1)Γ(s + 1) Γ(r + s + 2) a
1
≤ (b − a)(M − m)2 ,
4
where m ≤ f ≤ M a.e. on [a, b] and r, s ≥ 0.
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Proof. Barnett and Dragomir [3] established the following identity:
2 Z b
Z b
Z b
1
1
(4.6)
f (t)g(t)dt = p +
·
f (t) dt ·
g(t) dt,
b−a a
b−a
a
a
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where
1
|p| ≤ (Γ − γ)(Φ − φ), and Γ < f < γ, Φ < g < φ.
4
By taking g = f in (4.6), we get
Z b
1
(4.7)
f 2 (t)dt
b−a a
2
1
1
=p+
, where |p| ≤ (M − m), M < f < m.
b−a
4
Thus, (4.4) and (4.7) prove the theorem.
Another inequality based on a result from Barnett and Dragomir [3] follows:
Theorem 4.5. For the random variable X with distribution function F : [a, b] →
[0, 1],
Z b
Γ(r
+
1)Γ(s
+
1)
r
s
r+s
(4.8) (b − t) (t − a) f (t)dt − (b − a) ·
Γ(r + s + 2) a
1
≤ M (M − m)(b − a),
4
where m ≤ f ≤ M a.e. on [a, b] and r, s ≥ 0.
Proof. Barnett and Dragomir [3] have established the following inequality:
n Z b
1
1
n
(4.9) f (t)dt −
b−a a
b−a n−1
Γ2
Γ
· (b − a)n−1 − 1
≤
,
4(b − a)n−2
Γ · (b − a) − 1
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where γ < f < Γ.
From (4.9), we get
"
2 # 12
1 Z b
1
M
f 2 (t)dt −
, m ≤ f ≤ M.
≤
b − a a
b−a 2
and substituting in (4.4) proves the theorem.
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5.
Results Based on the Hölder’s Integral Inequality
We consider the Hölder’s integral inequality [4] and for t ∈ [a, b],
1, p > 1,
Z t
n
(n+1)
(t − u) f
(5.1)
(u)du
1
p
+
1
q
=
a
Z
≤
t
p1 Z t
1q
·
(t − u)nq du
|f (n+1) (u)|du
a
a
≤ ||f (n+1) ||p ·
nq+1
(t − a)
nq + 1
1q
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.
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On applying (5.1),we have the theorem:
Theorem 5.1. For the random variable X with distribution function F : [a, b] →
[0, 1], suppose that the density function f : [a, b] is n− times differentiable and
f (n) (n ≥ 0) is absolutely continuous on [a, b]. Then,
(5.2)
Z b
(t − a)r (b − t)s f (t)dt
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a
n
X
Γ(s
+
1)Γ(r
+
k
+
1)
−
(b − a)r+s+k+1 ·
Γ(r + s + k + 2) k=0
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
||f (n+1) ||∞

r+s+n+2 Γ(r + n + 2)Γ(s + 1)

·
(b
−
a)
·
,


n+1
Γ(r + s + n + 3)




if f (n+1) ∈ L∞ [a, b],








Γ(r + n + 1q + 1)Γ(s + 1)
||f (n+1) ||p

r+s+n+ 1q +1
1
· (b − a)
·
,
1
≤
·
(nq + 1)1/q
Γ(r
+
s
+
n
+
+
2)
q
n! 


if f (n+1) ∈ Lp [a, b] , p > 1,







Γ(r + n + 1)Γ(s + 1)



,
||f (n+1) ||1 · (b − a)r+s+n+1 ·


Γ(r
+
s
+
n
+
2)


if f (n+1) ∈ L1 [a, b],
where ||.||p (1 ≤ p ≤ ∞) are the Lebesgue norms on [a, b], i.e.,
Z b
p1
||g||∞ := ess sup |g(t), and ||g||p :=
|g(t)|p dt , (p ≥ 1).
t∈[a,b]
a
Proof. Using the Taylor’s expansion of f about a :
Z
n
X
(t − a)k k
1 t
f (t) =
f (a) +
(t − u)n f (n+1) (u)du, t ∈ [a, b],
k!
n!
a
k=0
we have
Z b
n Z b
X
f k (a)
r
s
r+k
s
(5.3)
(t − a) (b − t) f (t)dt =
(t − a) (b − t) dt ·
k!
a
a
k=0
Z b
Z t
1
r
s
n (n+1)
+
(t − a) (b − t)
(t − u) f
(u)du dt .
n! a
a
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Applying the transformation t = (1 − x)a + xb, we have
Z
(5.4)
b
(t − a)r+k (b − t)s dt = (b − a)r+s+k+1 ·
a
Γ(s + 1)Γ(r + k + 1)
.
Γ(r + s + k + 2)
For t ∈ [a, b],it may be seen that
Z t
Z t
n
(n+1)
(5.5) (t − u) f
(u)du ≤
|(t − u)n || f (n+1) (u)|du
a
a
Z t
(n+1)
≤ sup |f
(u)| ·
(t − u)n du
u∈[a,b]
≤ ||f
(n+1)
a
n+1
(t − a)
||∞ ·
n+1
Moments Inequalities of A
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Pranesh Kumar
.
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Further, for t ∈ [a, b],
Z t
Z t
n (n+1)
(5.6)
(u)du ≤
(t − u)n |f (n+1) (u)|du
(t − u) f
a
a
Z t
n
≤ (t − a)
|f (n+1) (u)|du
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a
≤ ||f (n+1) || · (t − a)n .
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Let
(5.7)
II
I
1
M (a, b) :=
n!
Z
a
b
(t − a)r (b − t)s
Z
a
t
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(t − u)n f (n+1) (u)du dt.
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Then (5.1) and (5.5) to (5.7) result in
(5.8) M (a, b)
 ||f (n+1) || R b
∞

· a (t − a)r+n+1 (b − t)s dt, if f (n+1) ∈ L∞ [a, b],

n+1



1  ||f (n+1) ||p R b
1
≤ · (nq+1)1/q · a (t − a)r+n+ q (b − t)s dt, if f (n+1) ∈ Lp [a, b] , p > 1,
n! 




Rb

||f (n+1) ||1 · a (t − a)r+n (b − t)s dt, if f (n+1) ∈ L1 [a, b].
Using (5.3), (5.4) and (5.8), we prove the theorem.
Moments Inequalities of A
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Pranesh Kumar
Corollary 5.2. Considering r = s = 1, the inequality (5.8) leads to
M (a, b)


||f (n+1) ||∞
(b − a)n+4


·
,
if f (n+1) ∈ L∞ [a, b],


(n + 1)
(n + 3)(n + 4)






1

(n+1)
||p
(b − a)n+ q +3
1  ||f
, if f (n+1) ∈ Lp [a, b] , p > 1, ,
≤
1 ·
1
1
q
 (nq + 1)
n! 
n+ q +2 n+ q +3








(b − a)n+3


,
if f (n+1) ∈ L1 [a, b],
 ||f (n+1) ||1 ·
(n + 2)(n + 3)
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which is Theorem 3 of Barnett and Dragomir [1].
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6.
Application to the Truncated Exponential Distribution
The truncated exponential distribution arises frequently in applications particularly in insurance contracts with caps and deductible and in the field of lifetesting. A random variable X with distribution function

 1 − e−λx for 0 ≤ x < c,
F (x) =

1
for x ≥ c,
is a truncated exponential distribution with parameters λ and c.
The density function for X :
 −λx
for 0 ≤ x < c
 λe
f (x) =
+ e−λc · δc (x),

0
for x ≥ c
where δc is the delta function at x = c. This distribution is therefore mixed with
a continuous distribution f (x) = λe−λx on the interval 0 ≤ x < c and a point
mass of size e−λc at x = c.
The moment generating function for the random variable X:
Z c
MX (t) =
etx · λe−λx dx + etc · e−λc
0

λ − te−c(λ−t)


, for t 6= λ,
λ−t
=



λc + 1,
for t = λ.
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For further calculations in what follows, we assume t 6= λ. From the moment
generating function MX (t), we have:
1 − e−λc
,
λ
2[1 − (1 + λc)e−λc ]
,
E(X 2 ) =
λ2
3[2 − (2 + 2λc + λ2 c2 )e−λc ]
E(X 3 ) =
,
λ3
4[6 − (6 + 6λc + 3λ2 c2 + λ3 c3 )e−λc ]
4
E(X ) =
.
λ4
E(X) =
Moments Inequalities of A
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Pranesh Kumar
The higher order central moments are:
Mk =
k X
k
i=0
i
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i
E(X ) · µ
k−i
, for k = 2, 3, 4, . . . ,
in particular,
1 − 2λce−λc − e−2λc
,
λ2
16 − 3e−λc (10 + 4λc + λ2 c2 ) + 6e−2λc (3 + λc) − 4e−3λc
=
,
λ3
65 − 4e−λc (32 + 15λc + 6λ2 c2 + λ3 c3 )
=
λ4
−2λc
3e
(30 + 16λc + 4λ2 c2 ) − 4e−3λc (8 + 3λc) + 5e−4λc
+
.
λ4
M2 =
M3
M4
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Using the moment-estimation inequality (3.6), the upper bound for M2 , in terms
of the parameters λ and c of the distribution:
M̂2 ≤
(1 − e−λc )(λc − 1 + e−λc )
.
λ2
The upper bounds for M3 using (3.8)
M̂3 ≤
(2 − 3λc + λ2 c2 ) − e−λc (6 − 6λc + λ2 c2 ) + 3e−2λc (2 − λc) − 2e−3λc
,
λ3
and using (3.9)
(−3 + 4λc − 3λ2 c2 + λ3 c3 )
M̂3 ≤
4λ3
e−λc (9 − 8λc + 3λ2 c2 ) − e−2λc (9 − 4λc) + 3e−3λc
+
.
4λ3
The upper bounds for M4 using (3.11)
M̂4 ≤
(−3 + 6λc − 4λ2 c2 + λ3 c3 ) + e−λc (12 − 18λc + 8λ2 c2 − λ3 c3 )
λ4
−2λc
2e
(9 + 9λc + 2λ2 c2 ) − 6e−3λc (2 − λc) + 3e−4λc
−
,
λ4
and from (3.12),
M̂4 ≤
(12 − 16λc + 103λ2 c2 − 4λ3 c3 + λ4 c4 )
4λ4
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4e−λc (12 − 12λc + 5λ2 c2 − λ3 c3 )
4λ4
−2λc
2e
(36 + 24λc + 5λ2 c2 ) − 16e−3λc (3 − λc) + 12e−4λc
+
.
4λ4
−
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References
[1] N.S.BARNETT AND S.S.DRAGOMIR, Some elementary inequalities for
the expectation and variance of a random variable whose pdf is defined
on a finite interval, RGMIA Res. Rep. Coll., 2(7) (1999). [ONLINE]
http://rgmia.vu.edu.au/v1n2.html
[2] N.S.BARNETT AND S.S.DRAGOMIR, Some further inequalities for univariate moments and some new ones for the covariance, RGMIA Res. Rep. Coll., 3(4) (2000). [ONLINE]
http://rgmia.vu.edu.au/v3n4.html
[3] N.S. BARNETT, P. CERONE, S.S. DRAGOMIR AND J. ROUMELIOTIS,
Some inequalities for the dispersion of a random variable whose pdf is defined on a finite interval, J. Ineq. Pure & Appl. Math., 2(1) (2001), 1–18.
[ONLINE] http://jipam.vu.edu.au/v2n1.html
[4] J.E. PEČARIĆ, F. PROSCHAN AND Y.L. TONG, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, 1992.
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