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ELSEVIER Computers
and Mathematics
with
Applications
47 (2004)
Journal
with applications 2336 www.elsevier.com/locate/camwa
Some Further Inequalities for Univariate Moments and Some New Ones for the Covariance N. S.
S. S.
BARNETT AND DRAGOMIR of Communications and Informatics Victoria University of Technology Box 14428, Melbourne City MC, Victoria 8001, Australia Qmatilda.vu.edu.au http://sci.vu.edu.au/staff/neilb.html http://rgmia.vu.edu.au/SSDragomirWeb.html School
P.O.
(Received
September
2002;
accepted
AbstractIn
this paper, some further inequalities ular reference to the expectations of the extremeorder obtained for the covariance of two continuous random reserved.
KeywordsGriks dom
variables,
inequality,
Chebychev’s
March
2003)
for univariate moments are given with particstatistics. In addition, some inequalities are variables. @ 2004 Elsevier Ltd. All rights
inequality,
Cumulative
distribution
functions,
Ran
Covariance.
1. INTRODUCTION The expectation of a continuous random variable is given by 00 E(X)
=
J m
xf (x) dx,
where f(z) is the probability density function of the random variable X. If the interval of definition is finite, (a, b), then this can be further expressed, using integration by parts, as b
E(X)
J
=b
a
F(x)
dx,
(1.1)
where F(z) is the associated cumulative distribution function. This result has been exploited variously to obtain inequalities involving the expectation and variance, see, for example, [l3]. The aim of this paper is to provide some additional inequalities utilising a generalisation of (1.1) to higher moments, as well as providing some specific results for the extremeorder statistics. In addition, some results are obtained involving the covariance of two random variables using a bivariate generalisation of (1.1) and generalisations of the inequalities of Griiss and Ostrowski.
08981221/04/$  see front matter doi: lO.lOlS/SO8981221(03)004486
@ 2004 Elsevier
Ltd.
All rights
reserved.
Typeset
by d&m
24
N. S. BARNETT
AND S. S. DRAGOMIR
2. UNIVARIATE Denote by M,, the nth moment s,” z”f(z)
RESULTS
d x which, using integration by parts, can be expressed
as
b
b”  n Consider now
1b IJ ba
gx
‘+lF
Ja
xnmlF(x) dx.
(2.1)
(x) dx  &[x”‘k+(x)
dj,
for which various inequalities can be found. Before exploiting difference in terms of the distributional moments, &[g”‘dxJo”F(T)
dx=
zbr;;
some of these, we express the
{bE(X)).
We therefore have 1 ba
b xnlF a
IJ
= bn n (b  a)
(x) dx  (b”  an) (b  E(X)) n (b  CA)~
MT% n (b  u)
= ab (anl  bnl)
b (bn  a”) + Ml (bn  an) n (b  a)’ n (b  a)2 M7l
(2.2)
+ Ml (b”  a”)
n (b  CL)~ iqq n (b  a)2 1 Ian (b  Ml)  b” (a  MI)  Mn (b  a)i. = n (b  a)2 Now utilising various inequalities, we can obtain a number of results. PREGR&s.
Using an inequality of [4] applied to (2.2), we have 1 lan (b  Ml)  bn (u  Ml)  M, (b  a)[ n (b  a)2
(2.3)
The special case when n = 2 gives Ia2 (b  E(X))
 b2 (a  E(X))
 (b  u) (CT”+ (E(X))‘)
1
< (b  CL)~[$$$
 (g$)2]‘;2>
that is, [(b  E(X))
(E(X)
 a)  0~1 5 (b  a)2
which is Theorem 3 of 131.
b2 + ab + a2 _ (b2 + 2ab + a”) 3 4
1 V2
(b  CL)~ =Q?
Some Further Inequalities
Using a further result of [4] and (2.2), we can obtain
PRECHEBYCHEV.
1
25
lun (b  kfl)
 bn (a  MI)
 M,, (b  a)1
n (b  a)2
112
1,
b @:a)
22(nl)dx
(&ibxn‘“‘“)2
s a
whereIlfllm is the supremum of the probability density function over (a, b), giving ian (b  MI) < VT(;;)’ 
,lf,,,
 b” (a  MI)
 A& (b  a)[ (2.4)
b2+l  a2+l _ (b  a) (2n  1)
The special case where n = 2 gives I(bE(X))(E(X)a)o’l~~llfll,, which was obtained by Barnett and Dragomir in [5].
3. LIPSCHITZIAN If s”lF(z)
MAPPINGS
is of the Lipschitzian type, then /xnlF
(x)  y”lF
(y)l 5 L Iz  y/ ,
where L 2 0 in which case X
“‘F(x)
 &
s”x+’ a
F(x)dx~5~[(~)2+(~)2](ba)
(Ostrowski’s inequality [S]). Now, s?lF
(x)  &
/“x+’ a
F(x)+xn‘F(~){~}~.
and thus, we have lx”‘F(x)
{$$$}I
5;
[(E)2+
(E)‘]
(ba).
If n = 2, then
Consider now the mapping F(s),
5 E [a, b], then the mapping is Lipschitzian
if there exists
L > 0 such that
IF (~1  F (Y)I I L Ix  YI Now, if F(.) is a cumulative distribution function, it is monotonic increasing between 0 and 1 over [a, b]. It is apparent that there exists .z E [x, y] such that
N. S. BARNETTAND S. ~.DFLAGOMIR
26
for z E [a,b], then this implies that F(x) is Lipschitzian, Thus, if we choose L = max[ y]2=, since IF (~1  F (~11I Ilfll, lx  ~1. Consider similarly the mapping OF, there exists z E [x, y] such that
XF (xl
it is also monotonic increasing and by the same token. YF (Y)
XY
In addition, we have that
and hence,
Thus, L can be taken to be
1+ Ml, max{lalTbl> and then
IxF (~1 YF(~11< [I+ Ilfll, maxU4, lb111 lx and so zF(x) is Lipschitzian. Similarly, it can be shown that x “lF(x)
YI 7
is Lipschitzian for n = 3,4,
[($+
(&)‘I (ba).
For x = a, we get p2
 b21 = 13+ ((E(X))“) b2/ 5 (b aj2P + Ilfll,maxC4 j PIN7
and, for x = b, we have (2b (b  u)  b2 + M2( = lb2  2ab + M,I = b (b  2a) + o2 + ((E (X))2)
)
i @ aI2P + llfll,max iI4 1PI>1 . 4. DISTRIBUTIONS MINIMUM AND
OF THE MAXIMUM, RANGE OF A SAMPLE
Consider a continuous random variable X with a nonzero probability density function over a finite interval [a, b]. Consider a random sample X1, X2, . . . , X,. We consider the distribution function of the maximum, minimum, and the range of the random sample. MAXIMUM. Let the cumulative distribution function of the maximum be G(x), the probability density function be g(x), and the corresponding functions for X be F(x) and f(x). Then
G(x)=Pr[max
and g(x) = n[F(s)]‘+‘f(a).
27
SomeFurther Inequalities MINIMUM. Let the cumulative distribution function of the minimum and the probability function be H(z) and h(z), respectively. Therefore,
density
and h (XT) = n [l  F (x)]“~
RANGE. Let the distribution be r(x). Consider
f(z)
function of the range be R(x)
and the probability
density function
K(s,t)=Pr[maxt] = [F(S)]”
 Pr {t 5 all Xl,.
= [F (s)]~  [F(s)
,X,
5 s}
 F (t)]”
Therefore, the joint probability density function of the extremeorder statistics can be found by differentiating this with respect to s and t, giving k (s, t) = 71(n  1) f (s) f (t) [F (s)  F (t)l”2
>
s > t,
which for a random variable defined on [a, b] gives
where 0 < x < b  a.
5. APPLICATION POSITIVE INTEGER As a preliminary
OF
GRtiSS’ POWERS
INEQUALITY TO OF A FUNCTION
to the proof of Griiss’ inequality we can establish the identity
ba1
J
bg(x)f(x) da:=p+ (+)‘i”m a
dx.[g(x) dX>
(5.1)
where it can be subsequently shown that
IPII a [r  71P  41> and y, P, 4, @ are, respectively, lower and upper bounds of f(z) and g(x). Applying this same identity to the square of a function, we have (5.2) Similarly, 1 J ba
b a
.f3(x)dx=p?+&{f(x) dx+Jbi2(2) a a dx
and using (5.1), we have 1 ba J
b a
f3 (x) dx =
p2
+ & J”a f (x)da:+
(5.3)
28
N.S.BAFLNETT AND S.S.DRAGOMIR
Continuing, we can show that for positive integers 72,TZ1 2
(5.4)
giving
where
bll L ,(W2, lP21 I ; (r  7)(6  $4,
Y< f (xl < l7 41< f2 (x)< 6,
lP31 I a (r  Y)(Q2 42), $2< f3 (xl < @2, Assuming that f(x) 2 0 and denoting ((l/(b  a)) s,” f(x) dx)n by Xn, we have
l&bfn@4 dx(A)’ ([f(x) dx)nI 5 a (r  71[ml  yl] + $ (r  7)[r+2  y2] x + i (r  7)[r+3  7+3]x2 +...
+ a (r  y)(r  7)~n~
Now, if f is a p.d.f., the righthand side reduces to
p1 (b_ a)n1 1 = 4p2 (b a)n2( r (b a) i ) r2 rn1(b ql  1 = 4 (b  ay2 ( ryba)1 1 .
(5.5)
\
Some
Further
Inequalities
29
If we now consider this inequality for an associated cumulative distribution have that A= bE(X) ba and the righthand side of (5.5) becomes 1 n1 4 CC i=l
bE(X) ba
[(b  a)+’
i1= >
function F(,)! we
 (b  E (x))“‘1
4(E(X)
a)(ba)n2
Thus, we have the two inequalities
1 b ba 1J a
r2 4 (b  a)n2
r1
(b  a)n1 _ 1 r (b  a)  1
Similarly, we can develop an inequality for 1  F(x) by suitable substitution x=
(5.6)
in (5.5), that is,
E(X)a
ba
’
which gives rl
b
& I
( J
b(lF(x))n dx& Ja 1 =b (1  F(x))” dx ba a
a
(1  F(x))
dx )I
I J
< [(b  a)n’  (E(X)  a)nl] 4 (b  a)n2 (b  E(X))
6. INEQUALITIES FOR THE OF THE EXTREMEORDER
(5.8)
’
EXPECTATION STATISTICS
As the p.d.f. of the maximum is
then
b
E [Km,]
Integrating
=
7~
by parts gives
J a
x [F (x)1”1 f (x) dx.
E[x,,,]= n [; [F(x)]f  ; J”a (F (x))~ b
=b
J a
giving, from (5.7),
F” (x) dx
dx
1
N. S. BARNETT
30
and when
E(X)
= (a + b)/2,
AND S. S. DRAGOMIR
we have
Wmxl &I, 1b E ba Consider
now E(X,ill)
= n s,“=, xf(x)[l
(““y).
 F(x)]“1
(6.2)
dx. Integration
by parts
gives
E [Xmin] = 72 [+F(x)]q+;lb(lF(x))Qx] and so
Ja
E(Xmin)=a+ Utilising
dx.
b(lF(~))n
(5.8), we have
and when
E(X)
= (a + b)/2,
we have (6.4)
7. APPLICATIONS The Beta probability
TO
density
function
x+l 1 B (a, P)
THE
is given
(1  x)pl
where
BETA
DISTRIBUTION
by
)
O
cr,p>o,
1
B(a,P)
=
J
2l
(1  x)‘l
dx.
0
Clearly,
E(X)
1
J
1 = B(a,p)
xa+ll
(1  x)o1
o
,jx = B(a+1 B (a, P)
l,P)
r (a!+ 1)r (a + p) CYr(a)r(a+P) = r(a+p+l)r(a) = (a+o)r(a:+p)r(a) Substituting
a = 0, b = 1, and letting
r E m, from
(5.6), we obtain
1
IS f”
dx
(x)

11 <
m2$:;;1i
,
0
and further, xn(l) I I&ii)
(1  x) +l)
dxl
I’
and m is the value of x for which (1  x)Ol
f’(x)
II
B (n (a  1) + 1, n (P  1) + 1) _ 1 Bn (a, P) < m2 (1  mnpl) 4(1m)
=
= 0, that is,
(o  1) x02
+ 2l
(p  1) (1  x)02
(1)
= 0,
Some Further Inequalities x:012 (1  .)P2
{(ck!  1) (1 x)
 Ic (p  1)))
i.e., Q1 (r+p2’
m=
a,p
> 1.
We then have the inequality
B(n(~l)+Ln(Pl)+l) Bn (a, P)
1
<  4(a+P2)(P1)
When Q = /?, the righthand Consider now (6.1) when
side becomes (l/8)(1  l/2”I). f(s) is the p.d.f. of a Beta distribution.
This gives
)l~E(X,.,)~(l~~)nJ=~l(~)~~~xmax)~~
‘l(:;;;:;;Jnl’
= (a+p)“’ Pl 4a (a + Py2 and when
’
Q = ,D, this becomes n
l
f
I
and, from
(6.3),
~E(X,i,)e and when
0
($)“i
’
[l  (cx/(a + LVnl] 4(1cu/(cr+P))
(a + p)n1  anl 4P(c1+/3)~~
=
’
CY= p
From these we can obtain
further
E(X,i,)
8. SOME BOUNDS PROBABILITIES INEQUALITIES
< l/2
and l/2
< E(Xmax)
< 3p
 (1/2)“1.
FOR JOINT MOMENTS AND USING OSTROWSKI TYPE FOR DOUBLE INTEGRALS
THEOREM 1. Let X, Y be two continuous random variables z E [a, b], y E [c, d] with probability density functions fl(.) and f2(.), respectively, and with joint probability density function f (., .) with associated cumulative distribution functions Fl(.), Fz(.), and F(., .). Then b
E(XY)=bE(Y)+dE(X)bd+ This is a generalisation
d
JJ.3=lX t=c
F (s, t) dsdt.
(8.1)
of the result b E(X)
F(s)
=bJ
and is equivalent
to b
E(XY)
ds
a
=bdd
d
Fl (s) ds  b Ja
b
d
JJ s=a
t=c
F(s,t)
F2 (t) dt + Jc
dsdt.
(8.2)
N. S. BARNETTAND S. %DRAGOMIR
32 PROOF.
and
' fWW];ca lla(J,,f("J)du) ds [/zl=lZ J.¶=a(Ju=af (u,t)du> ds,
b
J
sf(s,t)
ds =
s
a
b
=
bf2
@I
s

so
Now d
J t=c
tf (u,t) dt = [t Il,f
(“,v) d]f=c  L,
= dfi (u)  L,
(IL,f
(u7v)d”) dt
(I’=, f (~7v> dv) dt,
E (XV = bE (Y)  /I, (l;,
{ dfi (4  i/,
(I’=, f (WV) dv) dt} du) ds
b
J
=bE(Y)d
s=CZ b
d
JJ
=bE(Y)+dE(X)bd+
F(s,t)
*=a t=c
dsdt
and, equivalently, d
b
E(XY)
= bdd
J a
FI (s) ds  b
J c
bd
F2 (t) dt +
JJ a
F (s, t) ds dt.
c
I
In [‘i’], Barnett and Dragomir proved the following theorem. 2. Let f : [a,b] x [c, dj + lR be continuous (a, b) x (c, d) and is bounded, i.e.,
on [a, b] x [c, d],
THEOREM
llfilt
then we have the inequality
I/J b
a
c
for all (x, y) E [a, b] x [c, d].
Iloc) :=
sup (w)E(d)x(c,d)
a2f I
(GY)
asay
< I
o.
’
f&, = a
exists on
Some
Further
33
Inequalities
If we apply this taking f(., .) to be a joint cumulative distribution y = d, we obtain d
d
b
IJJ
F (s, t) ds dt  (b  u)
a
c
J
b
F (s, d) ds + (d  c) (b  a)
F (b, t) dt  (d  c)
c
function F(., .) with 2 = b,
That is. b
d
IJJ a
c
d
F (s, t) ds dt  (b  u)
J c
b
F2 (t) dt  (dc)
J
Fl (s) ds + (d  c) (b  u)
a
I $ (b aI2(d cl2IIF;;tIl, Using (8.2), this gives
~E(XY)+.[Fz(t) dt+cs,”
Fi(s)dsadbc+uc
1
=]E(XY)+uE(Y)cE(X)+uc]
I ; (b 4” Cd 4” IIF,‘:tllm = ; (b  u)~ (d  c)” Ilfll,
providing bounds for E(XY) Since Cov(X, Y) = E(XY)
,
in terms of E(X) and E(Y).  E(X) . E(Y), we can write the lefthand side alternatively Cov (X, Y) + [c  E(Y)]
[u  E(X)]
as
.
We can similarly extract other bounds from (8.3) in the situations where (i) IL:= b, y = c, (ii) z = a, y = d, and (iii) z = a, y = c, giving, respectively, IE (XY)
 dE (X)  UE (Y) + ud( 5 ; (b  u)~ (d  c)~ Ilfll,,
(E (XY)
 cE (X)
 bE (Y) + bc( 5 ; (b  a)’ (d  c)~ ilfll,,
IE (XV)
 dE (X)  bE (Y) + bdl 5 ; (b  u)~ (d  c)~ Ilfjl,
We can use the results of [8] by Dragomir, Cerone, Barnett and Roumeliotis to obtain further inequalities relating the first single and joint moments as well as some involving joint probabilities. In [8], bounds were obtained for
dsdt  , JJ f(s,t) b
d
fb,~)
(bi@

cl
a
c
N. S. BARNETTAND S. S.DRAGOMIR
34
namely, Ml(z) + A&(y) + M3(1c, y) where these are as defined in [8]. For one particular case we have
M2(y)= KW) Cd 4 + IY  cc+ 4Pll (b  u) (d  c) and
[O/2) (b 
M3 (x y) =
7
b)PllKW Cd 4 + IY  cc+ 4Pll @.f(s,t) (b u)(d c) II asat II1
a) + Ix  (a +
It follows then that if we choose f to be the joint cumulative distribution can generate the following inequalities: b
(b4b
4
d
JJ a
function, F(x, y), we
F (s, t) ds dt L Ml (a) +
M2
(c)
+ M3
(a, c)
c
and b
d
IJJ a
c
F(s,t)dsdtPr{XLz,Y
IM~(z)+Mz(y)+M3(x,y).
The first of these simplifies to give IE(XY)bE(Y)dE(X)+bdl<(ba)+(dc)+(dc)(ba).
9. FURTHER COVARIANCE
INEQUALITIES USING GROSS
FOR THE INEQUALITY
Consider functions fi(.), fi(.), and f(., .) w h ere f i is integrable over [a, b], f2 is integrable over [c, d] and f(x, y) is integrable for 2 E [a, b] and y E [c, d]. Consider the integral b
d
JJ a
fi
(3) f2
(t)
f (s, t)
f2
f (3, t)
ds dt
=
ds.
c
We have, for example,
Hence, b
d
JJ a
fi
(s)
(t)
ds dt
c
Cd 4+$ldf2(t) dt~df(sd) dt}ds afl(~1 =J {PI(s) b
=(d c)Jbn a (s)fi (3)ds+$%&t~d
(~f&)fWM) dt
Some
Further
Inequalities
35
and where
&AQ
J”
a
l1 (s)
Therefore, b
d
(t1.f(s>t) dsdt=(dc)Ja bagfl (s)ds+& JJ a c fl (s).fz
Jdfz(t)dt
d
Ji
X
(ba)PZw+&
u [fl
is~c~+~s.ti
ds]
dt
c
=(dc)Jbn a (s)fi (s)ds+$=+ Jd/2(t)dtJdm(t)dt + (b c&d  c)[h(t) d+liii dsl ~d;b,tldsdt, and thus, b
d
IJJ
fl (~1 f2 (t) f (s,t)
 (b  u)l(d  c) idic
2 (9 d(h
dsdt
(3) ds[ldf(6.t)
dsdj
5 (dc)IM, Jb vi(S)Ids+u a)IIP~II, Jd a c
~2
CASE 1. Now, if fl(s) lefthand side becomes E (XY)
= s and f2(t)
 4(dcj(~+
IPl
= t, and f(., .) is a joint
probability
[ !f;iTI/
 (&[tdt)
(d  c) (d2 + dc+
= ; If llm = i,,&
1’;’
(d”  c”)
x $
2 1’2
2 l/2
d2+t+c2
= ; Ilfll,
(see [4])
2 (d  c)
y
[
I [d2  2dc + c~]~‘~
= &
Ilfll,
Similarly, u2  b2 Pz<y7
a < 0,
b < 0.
Now ab If1 (s)l ds = T,
a < 0,
b > 0,
cdIfi (t)I dt L 7,
a > 0,
b > 0.
and similarly, J
density
( )I
c”) _
3 (d  c)
[
.I
(t11dt.
(d2cZ)(B2u2)~=~E(XY)~(b+u)(d+c)r.
(s)l I : llfllo.
(9.1)
(d  c)
function,
the
36
N. S. BARNETT
Thus,
we then
E(XY)
AND S. S. DFLAGOMIR
have, for a < 0, b > 0, c < 0, d > 0,
 a @+a)
(d+c)i
a2 + b2) (c” + d2) + (ac + db) (ad + bc)] Ilfll, CASE 2. If fl(s) density function, b
= s and f2(t) = 1, and f(.,.) then the lefthand side is
d
lJJ
stq5 (s, t) ds dt 
= tq5(s, t), where
qS(.,.) is a joint
. probability
(d  c) (b2 a”)
pa is as above and
Pl I ; Ilfll,
[$I,
($ldt)2]1’2=0,
and hence, E(XY)i(a+b)E(Y)
5 (b  aY$
+ d2) Ilfll,,
when a < 0, b > 0, c < 0, d > 0.
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