Journal of Inequalities in Pure and
Applied Mathematics
SOME INEQUALITIES BETWEEN MOMENTS OF PROBABILITY
DISTRIBUTIONS
volume 5, issue 4, article 86,
2004.
R. SHARMA, R.G. SHANDIL, S. DEVI AND M. DUTTA
Department of Mathematics
Himachal Pradesh University
Summer Hill, Shimla -171005, India
EMail: shandil_rg1@rediffmail.com
Received 23 February, 2004;
accepted 19 July, 2004.
Communicated by: T. Mills
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
040-04
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Abstract
In this paper inequalities between univariate moments are obtained when the
random variate, discrete or continuous, takes values on a finite interval. Further
some inequalities are given for the moments of bivariate distributions.
2000 Mathematics Subject Classification: 60E15, 26D15.
Key words: Random variate, Finite interval, Power means, Moments.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Some Elementary Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
Inequalities Between Moments . . . . . . . . . . . . . . . . . . . . . . . . . 9
4
Inequalities Between Moments of Bivariate Distributions . . . 14
5
Applications of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
References
Some Inequalities Between
Moments of Probability
Distributions
R. Sharma, R.G. Shandil, S. Devi
and M. Dutta
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1.
Introduction
The rth order moment µ0r of a continuous random variate which takes values on
the interval [a, b] with pdf φ(x) is defined as
Z b
0
(1.1)
µr =
xr φ(x)dx.
a
For a random variate which takes a discrete set of finite values xi (i = 1, 2,. . ., n)
with corresponding probabilities pi (i = 1, 2,. . ., n), we define
n
X
µ0r =
(1.2)
pi xri .
Some Inequalities Between
Moments of Probability
Distributions
R. Sharma, R.G. Shandil, S. Devi
and M. Dutta
i=1
The power mean of order r is defined as
1/r
Mr = (µ0r )
(1.3)
for r 6= 0,
Contents
and
1/r
Mr = lim (µ0r )
(1.4)
r→0
for r = 0.
It may be noted here that M−1 , M0 and M1 respectively define harmonic mean,
geometric mean and arithmetic mean.
Kapur [1] has reported the following bound for µ0r when µ0s is prescribed, r >
s, and the random variate, discrete or continuous, takes values in the interval
[a, b] with a ≥ 0,
(1.5)
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r/s
(µ0s )
(br − ar ) µ0s + ar bs − as br
≤ µ0r ≤
.
b s − as
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Inequality (1.5) gives the condition which the given moment values must
necessarily satisfy in order to be the moments of a probability distribution in the
given range [a, b]. Kapur [1] was motivated by the consideration of maximizing
the entropy function subject to certain constraints. But before maximizing the
entropy function one has to see whether the given moment values are consistent
or not i.e whether there is any probability distribution which corresponds to
the given values of moments. If there is no such distribution then the efforts
of finding out the maximum entropy probability distribution will not produce
any result and hence we should not proceed to apply Lagrange’s or any other
method to find the maximum entropy probability distribution, [2].
Here we try to obtain a generalization of inequality (1.5) for the case where r
and s can assume any real value. This shall help us in deducing bounds between
power means. This will also provide us with an alternate proof of inequality
(1.5) and enable us to tighten it when the random variate takes a finite set of
discrete values x1 , x2 ,. . ., xn .
In addition some inequalities between the moments of bivariate distributions
are also obtained.
Some Inequalities Between
Moments of Probability
Distributions
R. Sharma, R.G. Shandil, S. Devi
and M. Dutta
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2.
Some Elementary Inequalities
We prove the following theorems:
Theorem 2.1. If r is a positive real number and s is any non zero real number
with r > s then for a ≤ x ≤ b; with a > 0, we have
(2.1)
xr ≤
(br − ar ) xs + ar bs − as br
,
bs − as
and for x lying outside (a, b) we have
(2.2)
xr ≥
(br − ar ) xs + ar bs − as br
.
bs − as
If r is a negative real number with r > s then inequality (2.1) holds for x lying
outside (a, b) and inequality (2.2) holds for a ≤ x ≤ b.
Proof. Consider the following function f (x) for positive real values of x:
(2.3)
b r − ar s as b r − ar b s
x +
,
f (x) = x − s
b − as
b s − as
r
where r and s are real numbers such that r > s and s 6= 0. The function f (x) is
continuous in the interval [a, b] with a > 0. Then f 0 (x) is given by
r
b − ar
0
s−1
r−s
(2.4)
f (x) = x
rx − s s
.
b − as
Some Inequalities Between
Moments of Probability
Distributions
R. Sharma, R.G. Shandil, S. Devi
and M. Dutta
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f 0 (x) vanishes at x = 0 and c, where
1
r
r−s
s b − ar
(2.5)
c=
.
r b s − as
By Rolle’s theorem we have that c lies in the interval (a, b).
If r is a positive real number and s is a negative real number with r > s then
0
f (x) ≤ 0 iff x ≤ c. This means that f (x) decreases in the interval (0, c) and
increases in the interval (c, ∞). Further, since c lies in the interval (a, b) and
f (a) = f (b) = 0, it follows that
(2.6)
f (x) ≤ 0
for a ≤ x ≤ b,
R. Sharma, R.G. Shandil, S. Devi
and M. Dutta
and for x lying outside (a, b)
(2.7)
f (x) ≥ 0.
On substituting the value of f (x) from equation (2.3) in inequalities (2.6) and
(2.7), we obtain inequalities (2.1) and (2.2) respectively.
If r is a negative real number with r > s then f 0 (x) ≤ 0 iff x ≥ c. This
means that f (x) increases in the interval (0, c) and decreases in the interval
(c, ∞). Since c lies in the interval (a, b) and f (a) = f (b) = 0 it follows that
inequality (2.7) holds for a ≤ x ≤ b while inequality (2.6) holds for x lying
outside (a, b) and thus we get inequalities for the case when r is negative real
number.
Theorem 2.2. For a ≤ x ≤ b with a > 0, we have
(2.8)
Some Inequalities Between
Moments of Probability
Distributions
(br − ar ) log x + ar log b − br log a
xr ≤
,
log b − log a
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and for x lying outside (a, b), we have
(2.9)
xr ≥
(br − ar ) log x + ar log b − br log a
,
log b − log a
where r is a real number.
Proof. Consider the following function f (x) defined for positive real values of
x,
(br − ar )
br log a − ar log b
f (x) = x −
log x +
.
log b − log a
log b − log a
Some Inequalities Between
Moments of Probability
Distributions
The function f (x) is continuous in the interval [a, b] where a > 0. Then f 0 (x)
is given by
1
b r − ar
0
r
(2.11)
f (x) =
rx −
,
x
log b − log a
R. Sharma, R.G. Shandil, S. Devi
and M. Dutta
(2.10)
r
and we have f 0 (x) = 0 at x = c where
r1
b r − ar
.
(2.12)
c=
r (log b − log a)
By Rolle’s Theorem we have that c lies in the interval (a, b). Also f 0 (x) ≤ 0 iff
x ≤ c. This means that f (x) decreases in the interval (0, c) and increases in the
interval (c, ∞). Further, since c lies in the interval (a, b) and f (a) = f (b) = 0
it follows that
(2.13)
f (x) ≤ 0
for a ≤ x ≤ b,
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and for x lying outside (a, b) we have
(2.14)
f (x) ≥ 0.
On substituting the value of f (x) from equation (2.10) in inequalities (2.13) and
(2.14), we obtain inequalities (2.8) and (2.9) respectively.
Some Inequalities Between
Moments of Probability
Distributions
R. Sharma, R.G. Shandil, S. Devi
and M. Dutta
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3.
Inequalities Between Moments
Theorem 3.1. Let r be a positive real number and s be any non zero real number with r > s. If a positive random variate takes values xi (i = 1, 2,. . ., n) in
the interval [a, b], with a > 0, then we have
(3.1)
µ0r ≤
(br − ar ) µ0s + ar bs − as br
,
b s − as
and
xrj − xrj−1 µ0s + xrj−1 xsj − xsj−1 xrj
,
xsj − xsj−1
(3.2)
µ0r ≥
where j = 2, 3,. . ., n.
If a continuous random variate takes values in the interval [a, b], with a > 0,
then the upper bound for µ0r is given by the inequality (3.1) whereas the lower
bound is given by following inequality
(3.3)
r/s
µ0r ≥ (µ0s )
.
Proof. It is seen that µ0r can be expressed in terms of µ0s in the following form :
!
r
r
x
−
x
xsβ xrα − xsα xrβ
α
β
0
(3.4) µ0r =
µ
+
s
xsβ − xsα
xsβ − xsα
"
#
n
r
r
r s
r s
X
x
−
x
x
x
−
x
x
α s
α β
β
β α
+
x +
,
pi xri − s
s
s i
s
x
−
x
x
−
x
α
α
β
β
i=1
Some Inequalities Between
Moments of Probability
Distributions
R. Sharma, R.G. Shandil, S. Devi
and M. Dutta
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where α and β take one of the values among 1, 2,. . ., n with α 6= β. Without
loss of generality we can arrange values of the variate such that a = x1 ≤
x2 ≤ · · · ≤ xn = b. If we take α = 1 and β = n then x1 ≤ xi ≤ xn
for i = 1, 2, ,. . ., n. It follows from (2.1) that the last term in equation (3.4)
is negative and we conclude that the upper bound for µ0r is given by inequality
(3.1). Further if xα = xj−1 and xβ = xj , j = 2, 3,. . ., n then each xi lies outside
(xj−1 , xj ) and it follows from (2.2) that the last term in equation (3.4) is positive
and we conclude that the lower bound for µ0r is given by inequality (3.2). It is
also clear that equality in the inequalities (3.1) and (3.2) holds iff n = 2.
If the value of µ0s coincides with one of xsj−1 or xsj , then from inequality (3.2)
we have
µ0r
(3.5)
≥
r/s
(µ0s )
.
Also if xj−1 approaches xj we get inequality (3.5) and we conclude that for a
continuous random variate the lower bound for µ0r is given by inequality (3.5).
The upper bound for µ0r can be deduced from Theorem 2.1. Multiplying both
sides of inequality (2.1) by pdf φ(x) we get, on using the properties of definite
integrals, inequality (3.1).
Theorem 3.2. Let r and s be negative real numbers with r > s. If a positive
random variate takes values xi (i = 1, 2,. . ., n) in the interval [a, b], with a > 0,
we have
(3.6)
µ0r
(br − ar ) µ0s + ar bs − as br
,
≥
b s − as
Some Inequalities Between
Moments of Probability
Distributions
R. Sharma, R.G. Shandil, S. Devi
and M. Dutta
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and
(3.7)
0
r
r
x
µs + xrj−1 xsj − xsj−1 xrj
−
x
j
j−1
0
,
µr ≤
xsj − xsj−1
where j = 2, 3,. . ., n.
If a continuous random variate takes values in the interval [a, b], with a > 0,
the lower bound for µ0r is given by inequality (3.6) whereas the upper bound for
µ0r is given by following inequality:
(3.8)
µ0r ≤
r/s
(µ0s )
.
Proof. We again consider equation (3.4). If we take α = 1 and β = n then
x1 ≤ xi ≤ xn for i = 1, 2,. . ., n. It follows from Theorem 2.1 that the last
term in equation (3.4) is positive and we conclude that the lower bound for µ0r
is given by inequality (3.6). Also if xα = xj−1 and xβ = xj , j = 2, 3,. . ., n
then each xi lies outside (xj−1 , xj ). It follows from Theorem 2.1 that the last
term in equation (3.4) is negative and we conclude that the upper bound for µ0r
is given by inequality (3.7). Also if xj−1 approaches xj we get inequality (3.8).
The lower bound for µ0r can be deduced from Theorem 2.1. Multiplying both
sides of inequality (2.2) by pdf φ(x) we get, on using the properties of definite
integrals, inequality (3.6).
Some Inequalities Between
Moments of Probability
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R. Sharma, R.G. Shandil, S. Devi
and M. Dutta
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Theorem 3.3. For a random variate which takes values xi (i = 1, 2,. . ., n) in
the interval [a, b], with a > 0, we have
(3.9)
(br − ar ) log M0 + ar log b − br log a
,
µ0r ≤
log b − log a
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and
(3.10)
r
r
r
r
x
−
x
j
j−1 log M0 + xj−1 log xj − xj log xj−1
µ0r ≥
,
log xj − log xj−1
where j = 2, 3,. . .n, r is a real number and
M0 = xP1 1 xP2 2 · · · xPnn .
(3.11)
For a continuous random variate which takes values in the interval [a, b] with
a > 0 the upper bound for µ0r is given by inequality (3.9) whereas the lower
bound for µ0r is given by the following inequality
µ0r
(3.12)
r
≥ (M0 ) .
Proof. It is seen that µ0r can be expressed in terms of log M0 in the following
form:
xrβ − xrα
xrα log xβ − xrβ log xα
log M0 +
log xβ − log xα
log xβ − log xα
n
r
r
X
xrβ log xα − xrα log xβ
xβ − xα
r
log xi +
.
+
Pi x i −
log xβ − log xα
log xβ − log xα
i=1
(3.13) µ0r =
Without loss of generality we can arrange values of the variate such that a =
x1 < x2 < · · · < xn = b. If we take α = 1and β = n then x1 ≤ xi ≤ xn
for i = 1, 2,. . ., n. It follows from Theorem 2.2 that last term in equation (3.13)
is negative and we conclude that the upper bound for µ0r is given by inequality
Some Inequalities Between
Moments of Probability
Distributions
R. Sharma, R.G. Shandil, S. Devi
and M. Dutta
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(3.9). Also if xα = xj−1 and xβ = xj , j = 2, 3,. . ., n then each xi lies outside
(xj−1 , xj ). It follows from Theorem 2.2 that the last term in equation (3.13)
is positive and we conclude that the lower bound for µ0r is given by inequality
(3.10).
If the value of M0 coincides with one of xj−1 or xj then from inequality
(3.10) we have
(3.14)
µ0r ≥ (M0 )r .
Also if xj−1 approaches xj we get inequality (3.14) and we conclude that for the
continuous random variate the lower bound for µ0r is given by inequality (3.14).
The upper bound for µ0r can be deduced from Theorem 2.2. Multiplying both
sides of inequality (2.8) by pdf φ(x) we get, on using the properties of definite
integrals, inequality (3.9).
Some Inequalities Between
Moments of Probability
Distributions
R. Sharma, R.G. Shandil, S. Devi
and M. Dutta
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4.
Inequalities Between Moments of Bivariate
Distributions
The moments of a bivariate probability distribution are the generalizations of
those of univariate one and are equally important in the theory of mathematical
statistics. For a discrete probability distribution, if pi is the probability of the
occurrence of the pair of values (xi , yi ) i = 1, 2,. . ., n, the moment µ0rs about
the origin is given by
µ0rs
(4.1)
=
n
X
Pi xri yis .
i=1
We obtain a bound on µ0rs in the following theorem:
Theorem 4.1. Let µ0rs be the moment of order r in x and of order s in y, about
the origin (0, 0), of a discrete bivariate probability distribution. The random
variates x and y vary respectively over the finite positive real intervals [a, b]
and [c, d]. If µ0k m is the corresponding moment of order k in x and m in y such
that r ≥ k, s ≥ m and rm = ks then we must have by necessity,
r+s
(4.2)
(µ0k m ) k+m ≤ µ0r s ≤
(br ds − ar cs ) µ0k m + ar cs bk dm − ak cm br ds
.
bk dm − ak cm
Proof. If u, v, α and β are positive real numbers with α + β = 1 then from
Hölder’s inequality [3],
!α
!β
n
n
n
X
X
X
α β
(4.3)
ui v i ≤
ui
vi .
i=1
i=1
i=1
Some Inequalities Between
Moments of Probability
Distributions
R. Sharma, R.G. Shandil, S. Devi
and M. Dutta
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We make the following substitutions,
(4.4)
ui = pi xri yis , vi = pi and α =
k+m
.
r+s
This gives,
uαi viβ = pi xki yim .
(4.5)
Also,
(4.6)
n
X
!α
ui
=
n
X
i=1
Some Inequalities Between
Moments of Probability
Distributions
! k+m
r+s
pi xri yis
,
i=1
R. Sharma, R.G. Shandil, S. Devi
and M. Dutta
and
(4.7)
n
X
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!β
vi
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= 1.
i=1
JJ
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From (4.3), (4.5), (4.6) and (4.7), we get
(4.8)
µ0r s ≥ (µ0k m )
r+s
k+m
.
For a ≤ x ≤ b, c ≤ y ≤ d, r ≥ k, s ≥ m and rm = ks, inequality (4.3)
will remain valid if we substitute n = 2, u1 = p1 ar cs , u2 = p2 br ds , v1 = p1 ,
v2 = p2 , α = k+m
,
r+s
bk dm − xk y m
p1 = k m
,
b d − ak c m
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and
p2 =
x k y m − ak c m
.
bk dm − ak cm
These substitutions give
(4.9)
xr y s ≤
(br ds − ar cs ) xk y m + ar cs bk dm − ak cm br ds
.
bk dm − ak cm
Without loss of generality we can have that the random variate take values a =
x1 < x2 < · · · < xn = b and c = y1 < y2 < · · · < yn = d therefore a ≤ xi ≤ b
and c ≤ yi ≤ d, i = 1, 2,. . ., n. From inequality (4.9), it follows that
xri yis ≤
or
n
X
Pi xri yis ≤
(br ds − ar cs ) xki yim + ar cs bk dm − ak cm br ds
,
bk dm − ak cm
(br ds − ar cs )
i=1
Pn
r s k m
k m r s
k m
i=1 Pi xi yi + a c b d − a c b d
bk dm − ak cm
or
R. Sharma, R.G. Shandil, S. Devi
and M. Dutta
Pn
i=1
Pi
Title Page
,
(br ds − ar cs ) µ0k m + ar cs bk dm − ak cm br ds
.
µ0r s ≤
bk dm − ak cm
Inequality (4.2) also holds for the continuous bivariate distributions. The upper bound in inequality (4.2) is a consequence of inequality (4.9). Multiplying
both sides of inequality (4.9) by joint pdf φ(x, y) and integrating over the corresponding limits, we get the maximum value of µ0rs where
Z bZ d
Z bZ d
0
r s
µrs =
x y φ(x, y)dxdy
and
φ(x, y)dxdy = 1.
a
c
a
c
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Now consider,
RbRd
f α g β dxdy
α R R
β
b d
f dxdy
g dxdy
a c
a
R R
b d
a
c
c
Z bZ
d
=
a
≤
a
RbRd
c
Z bZ
c
a
d
!α
f
"
c
!β
g
RbRd
f dxdy
a
c
dxdy
g dxdy
#
αf
βg
+ RbRd
dx dy
RbRd
f
dxdy
g
dxdy
a c
a c
Some Inequalities Between
Moments of Probability
Distributions
= 1,
R. Sharma, R.G. Shandil, S. Devi
and M. Dutta
where α + β = 1 and f and g are positive functions. We therefore have
Z bZ
d
α β
Z b Z
f g dx dy ≤
(4.10)
a
c
α Z b Z
d
f dxdy
a
c
g dxdy
a
φ(x, y), g = φ(x, y)
,
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c
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J
and make the following substitutions,
f = xr y s
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β
d
and
Inequality (4.10) then yields the minimum value of µ0rs .
α=
k+m
.
r+s
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5.
Applications of Results
On using the results derived in Section 3 and giving particular values to r and
s it is possible to derive a host of results connecting the Harmonic mean (H),
Geometric mean (G), Arithmetic mean (A) and Root mean square (R) when one
of the means is given and the random variate takes the prescribed set of positive
values x1 , x2 , . . . , xn .
If we put r = +1 and s = −1 we get inequalities between A and H, if we
put r = 0 and s = −1 we get inequalities between G and H, and so on. Root
mean square R corresponds to r = 2. In particular the following inequalities
are obtained from the general result,
1
1
(5.1)
[(xj−1 + xj ) A − xj−1 xj ] 2 ≤ R ≤ [(a + b) A − ab] 2 ,
(5.2)
ab
xj−1 xj
≤H≤
,
a+b−A
xj−1 + xj − A
(5.3)
bA−a ab−A
1
b−a
A−xj−1 xj −A
xj−1
≤ G ≤ xj
1
j −xj−1
x
Some Inequalities Between
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(5.4)
1
xj−1 xj (xj−1 + xj ) 2
2
2
xj−1 + xj−1 xj + xj −
H
1
ab (a + b) 2
2
2
≤ R ≤ a + ab + b −
,
H
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(xj−1 + xj ) −
(5.5)
(5.6)
h
x (H−xj−1 ) xj−1 (xj −H)
xj j
xj−1
log
(5.7)
(5.8)
G
xj−1
x
ab
log ab
a b b
log Ga
G
G
xj
xj−1
log
1
j −xj−1 )
i H(x
xj x2j−1
G
j
log xj−1
log
(5.9)
x2j
ab
xj−1 xj
≤A≤a+b− ,
H
H
xj
xj−1
1
≤ G ≤ bb(H−a) aa(b−H) H(b−a) ,
log
≤ R2 ≤
2
G b
a
2
b a
G
log ab
xj−1 xj
xj
log xj−1
xj−1
xj xj
G
log xj−1
G
xj xj−1
G
≤A≤
log
G b
a
log ab
b a
G
,
R. Sharma, R.G. Shandil, S. Devi
and M. Dutta
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≤H≤
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(5.10)
R2 + ab
R2 + xj−1 xj
≤A≤
,
a+b
xj−1 + xj
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J. Ineq. Pure and Appl. Math. 5(4) Art. 86, 2004
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(5.11)
a2
ab (a + b)
xj−1 xj (xj−1 + xj )
≤H≤ 2
,
2
2
+ ab + b − R
xj−1 + xj−1 xj + x2j − R2
and
(5.12)
b
R2 −a2
b2 −a2
a
b2 −R2
b2 −a2
R2 −x2
j−1
x2 −x2
j
j−1
≤ G ≤ xj
2
x2
j −R
2
2
x −x
j
j−1
xj−1
,
where j = 2, 3,. . ., n.
We now deduce the result that the power mean Mr is an increasing function
of r. If r is positive and s is any real number with r > s then from inequality
(3.3) we have
1
(5.13)
1
(µ0r ) r ≥ (µ0s ) s ,
or Mr ≥ Ms.
If r is a negative real number with r > s we again get inequality (5.13) from
inequality (3.8). From inequality (3.12) we have Mr ≥ M0 for r > 0, and
Mr ≤ M0 for r < 0. Hence we conclude that the power mean of order r is an
increasing function of r. In particular, we get that
Some Inequalities Between
Moments of Probability
Distributions
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and M. Dutta
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M−1 ≤ M0 ≤ M1 ≤ M2 .
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References
[1] J.N. KAPUR AND A. RANI, Testing the consistency of given values of a
set of moments of a probability distribution, J. Bihar Math. Soc., 16 (1995),
51–63.
[2] J.N. KAPUR, Maximum Entropy Models in Science and Engineering, Wiley
Eastern and John Wiley, 2nd Edition, 1993.
[3] G.H. HARDY, J.E. LITTLEWOOD AND G. POLYA, Inequalities, Cambridge University Press, 2nd Edition, 1952.
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and M. Dutta
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