Gcrtrude M.
•
ROUNDS FOR THE
D'r:~:Tl~I'3UTION
FUNCTION
or
A
Sm·I
OF INDEPENI:lENT,
ID~NTICA.LLY DISTRIBUTED EANDQM VARIABLES1
by
.Wassily Hoeffding and S. S. Shrikhande
University of North Carolina and Universitv of Nagpur
Institute of Statistics
Mimeograph Series No. 84
Limited Distribution
October, 1953
1This research was supported by the United States Air Force under
Contract No. AF 18(600)-458 monitored by Headquarters, Air Research and
Development Command•
•
COA
Bounds for the distribution function of a Sl~ of independent,
identically distributed random variables.~~
ByWassily Hoeffding and S. S. Shrikhande,
Uni"\'ersity of North Carolina and University of Nagpur
SW®lary.
The problem is considered of obtaining bounds for the
cumulative distribution function of the sum of n independent, identically
distributed random variables with k prescribed moments and given range.
For n
=2
c10sely
it is shown that the best bounds are attained or arbitrarily
~pproached
2k + 2 values.
1..Jith discrete random varia')les which take on at most
Explicit bounds are obtained for the case of nonnegative
random varia 1'les with given mean when n
=
2; for arbitrary values of n
bounds are given \-Thich are asymptotically best in the "tail" of the distribution.
Some of the results contribute to the more general problem of
obtaining bounds for the expected value of a given function of independent,
identically distributed random variables when the expected values of
certain functions of the individual variables are Given.
1.
Introduction.
The problem considered in this paper is part of
the followin8 general problem.
Let D be the class of all cdfs (cumulative
=
distribution functions) F(x) on the real line Hhich satisfy the conditions
~~This research was supported by the United States Air Force under
Contract No. AF 18(600)-458 monitored by Headquarters, Air Research and
Development Comruand.
- 2 -
(1.1)
i = 1, ••• , k,
(1.2)
F(x)
=
° if x
<
F(x)
A,
= 1 if x
>
B ,
where the_functions gl(x), ••• , gk(X) and the constants c ' ••• , c ' A, B
l
k
are given. We allow that A = - 00 and/or B = 00. Here and in what follows,
when the domain of integration is not indicated, the integral extends over
the entire range of the variables involved.
Let K(x l , ••• , x n ) be a function such that
...
exists for all F in D in the sense that the multiple integral is equal to
the repeated integral taken in an arbitrary order.
termine upper and lOl-JCr bounds for
For n
1/1
The problem is to de-
(F) when F is in D.
=
= 1, gi(x) = xi, K(x) = 1 or 0 according as x ~ t or
> t,
as well as for other functions K(x), an extensive literature on the subject exists, associated with the names of
Diena~ne,
Chebyshev, Markov,
L- 6_7.
by Wald L- 7 _7.
Stieltjes, and others; for references see Shohat and Tamarkin
case n -- 1, g.(x) = x ~ ,
A = 0, B =
00
For n arbitrary, Robbins ~ 5
_7
J.
was treated
The
showed that the Biena~~ - Chebyshev
- 3 bound for Prob.
(Ix l
+ ••• + X
n
I~
t), where the Xi are independent and
identically distributed with zero mean and given variance, can be improved
when n > 1.
Plackett
L- 4 _7
and Hartley and David
L- 2 _7
obtained the
best possible bounds for the expected sample range and the expected value
of the largest observation in the case where the mean and the variance are
given, assuming that the common cdf is continuous.
A problem analogous
to the general problem stated above, but without the assumption that the
n variables are identically distributed, was considered by one of the
L- 3 _7 who
authors
showed that under general conditions the best bounds
are attained or arbitrarily closely approached with step functions in D
=
which have at most k + 1 steps.
In the present paper the attention is concentrated on the case where
k ; 1 or 0 according as a given function f(x , ••• , x ) is or is not conl
n
tained in a given set. The method used permits to obtain the closest bounds
only for n ; 2.
for n
=
If f ; xl + ~ •• + x n ' n is even, and gi(x)
= xi,
the bounds
2 can be applied in an obvious way, but will not in general be the
best ones.
]\fore general functions K are considered only insofar a s they
can be handled by the same method.
Theorem 2.1 states conditions under which we need consider only
step functions in D.
of a certain
Theorems 2.2 and 2.3 show that for functions K(x,y)
=
type we may restrict
bounded number of steps.
our attention to step functions with a
In Theorem 3.1 an explicit expression for the
least upper bound of Prob. (X + Y ~ t) is obtained when X and Yare independent, identically distributed and nonnegative, with given mean.
In
-4section
4 bounds
2.
for the analogous case with n smrunands are considered.
The least upper bound of
j)CK(X,y) dF(x) dF(y).
Let D be
=
the class of cdfs F(x) which satisfy the conditions (1.1) and (1.2).
Let
K(x,y) be a function such that
exists for all F in
£'
in the sense that the double integral equals
the repeated integral.
the greatest lower
ff K(x,y) dF(x) dF(y)
=
\jr (F)
The orob1em is to determine the least upper and
bO~Ld
of\jr(F) for all F
in~.
As long as the function
K(x,y) is not specified it is sufficient to consider the least upper
bound.
1f.
Let D be the class of all F in D ~ich
=
=
finite number of steps.
~re
step functions with a
The following obvious theorem gives sufficient
conditions for confining our attention to functions in D·:~•
-
.
Theorem 2, L
A)
Supoose that
~f-
?~
for every F in D and every 6 > 0 there exists an F in D
----
-=
---
such that
sup
x
B)
for
~
F
~
I F1~(X)
- F(x)
g and every
6
that for any G in D for which
-=
I
< 6
,.
> 0 there exists a 5 > 0 such
-5-
I F(x)
sup
x
- G(x)
I
I<
e.
< 0
we have
I W(F)
.. 1jr(G)
Then
sup '!reF)
FeD
= SUP"L
FeD'~
'!r(F).
=
=
The proof is omitted.
In.
Assumption A) is satisfied if gi(X) = x
-
J.,
where ml""'~ are
arbitrary integers, that is, if D is the class of cdfs with k prescribed
moments and given range.
This follows from Lemma 3.1 in L:3
_7.
A particular case in which assumution B) is satisfied is where
K(x,y)
= 1 or
0 aC00rding as (x,y) is or is not contained in a Borel
set S with the property that the sets of all points x such that (x,y) e S
and the sets of all y such that (x,y) e S are unions of a finite and
bounded number of intervals.
This is a special case of Theorem
4.1
in
We shall now derive sufficient conditons under which, given a
step function F in
Q with
m steps, we can construct a step function G in
D with less than m steps such that 'irCG)
=
~
1lr(F).
- 6 A step function F in D with exactly m steps is of the form
=
F(x)
= P.
J
J.'f aJ'
_< x < a + ,
j
J=
. 0 , 1 , •••
1
,m,
where
a
a
m+l
= 00
,
< B ,
m-
(2.4)
m
Z g.(a.)(P. - P. 1) = c, ,
1
1
J
J
J-
j=l
i = 1, ••• , k •
If we define
the last equations can be written
i = 1, • II' k
-7Let
1'f
j=O,l, ••• ,m.
a. < x < a. l'
J+ •
J -
In order that G(x) be a cdf in D i t is sufficient that the numbers t, D.
=
J
satisfy the conditions
°
DO
= Dm
='
(2.8)
m-1
.6 h .. D. = 0,
j=l 1J J
i=l, ••• ,k.
If F and G are defined by (2.2) and (2.7), we have
m
(2.11)
1Jr
(F)"'.6
i=l
m
.6
j=l
K.. (P. - P. l)(P, - P. 1)'
1J
1
1-
where
(2.12)
K.. = K(a., a.),
1J
1
J
J
J-
-8and
m-1
2 m-1 m-1
(2.13) *(G) - *(F) = t Z L.D. + t
Z
Z L.. D.D.,
j=l J J
i=l j=l ~J ~ J
where
m
L. = Z (K.. + K . -l
J•
J
i=l ~J
Lemma 2.1.
~
defined by (2.2)
u1'.'.'~+1
K.
. +1 - K 'L1 .) ( p. - P. 1)'
~,J
JT,~
~
~-
F be a step function in D with exactly m steps,
-Let
-
-
=------:..
(2.5), where m> k+1.
Suppose that the integers
can be so chosen that
1 :;
~ < u
2 < •••
<: ~+ 1
:; m-1
and the equations
k+1
(2.16)
Z
r=l
h.
~u
r
x
r
= 0,
i
= 1, ••• ,
k,
-9-
k+l k+l
~
~
r=l 8=1
L
x x
uu
>
rs-
r s
O.
Then there exists a step function G in D with less than m steps, for
which 1jF (G) ~ 1jF( F) "
Proof.
Let G( x) be defined by (2.7), and let
D.
J
Let A
=1
=0
for j
r u1 ,···, U,.<:+ I"
or 0 according as the rank of the matrix
·..
.....
·..
·" .
Then the equations (2.16) and
is equal to or less than k+l.
k+l
~
r=l
L
u
x
r
r
=
-10-
have a solution (D
ul
, ••• , D
)
'1<+1
I
(0, ••• , 0).
Having thus fixed the
D., let t be the largest number which satisfies the inequalities (2.9).
J
This number exists and is positive.
With this choice of the numbers t
and D., G is a step function in D with less than m steps.
=
J
Furthermore,
by (2.13),
W(G) -W(F)
=
U. + t
2
k+l k+ 1
~
1:
r=l s=l
L
uu
r s
D D > O.
u u
r s
The proof is complete.
The next theorem shows that if K(x,y) is of a certain form, and if
we restrict ourselves to the class D'~ of step functions in D with a finite
==
::
number of steps, we need consider only step functions with a bounded
number of steps.
Let D be the class of all F in D which are step functions with
=
=m
at most m steps.
Theorem 2.2.
Suppose that K(x,y) is of the form
k
K(x,y) = ~
k
~ a~ .. g1.'(x) gJ'(y)
i=O j=O '\l1.J
-11-
t=l, ••• ,s,
where gO(x) = 1, the a
-00
~
tij
are arbitrary constants,
= bo < b1 <... < b s-1 < b s =
00,
f(x,y) is a strictly increasing function in each of its arguments
when the other argmuent is fixed.
Then
sup" W(F)
sup
FeD"
FeD
=
W(F).
=sk+s
The theorom romains true if in the inequalities b _
t l
some signs
~
~
f(x,y) < b
t
are replaced by < or vice versa, provided that the s sets
defined by the inequalities cover the entire plane.
Proof.
Let F(x), as defined by (2.2) to (2.5), be an arbitrary
step function in D with exactly m steps, where m > sk + s.
=
It is
sufficient to construct a step function G in D with less than m steps
such that W(G)
~
W(F).
=
Let m (t=l, ••• ,s) denote the number of indices
t
u (1 ~ u ~ m) for "'Thich
b t 1 < f(a , a ) < b t •
- u
u
-12Then
s max (mt )
~ ~ + •••
+m
s
=
m > s(k+l).
Hence thero exists a t for vnich mt
~
k
+ 2
and an integer n such that
b t - 1 -< f(a n , a n ) < f(a n+ k+l' a
l ) < bt •
n+lc+
The assumption about f(x,y) implies that
Kvw
~
k
k
~
~
i=O j=O
at,. g.(a ) g.(a )
lJ
l
V
J
for n
~
v, w < n+k+l.
W
By (2.15) and (2.6) this implies
L
vw
=
k
~ a
, h. h.
i=l j=l tiJ lV JW
Honce if we let ur
are satisfied.
_.
If g.(x)
l
k
~
for n
~
v,
W~
n+k.
= n+r-l, r=1,2, ••• ,k+l, tho conditions of Lemma 2.1
The proof is complete.
= xi,
that is, if D is the class of distributions with
=
-13given moments up to order k and giv(m range, the assumption of Theorem 2.2
means that K(x,y) is piecewise polynomial, of bounded degrees, in sections
of the plano separated by curves of negative slope.
note that i f K(x,y) is piecewise
curves of positive slope, a
polyno~tal in
si~Llar
It is of interest to
sections separated by
reduction of the problem to the case
of step functions with a boundod number of steps is in general
imnossible.
For examnle, let
K(x,y)
= max (x,;y·),
and let D be the class of cdfs F with
=
! x 2 dF(x)
J x dF(x) = 0,
=
1.
Under the restriction to continuous functions F(x) this is a special case
of a problem considered by Hartley and David
L-2_7.
cdf F(x) we can write
if (F)
2/
x F(x) dF(x),
where
F(x)
= (F(x-0)+F(x+0))/2.
For an arbitrary
-14Using Schwarz's inoquality, we have for any constant c and any F in ::::D
J (x+c) F(x) dF(x)
1jr(F) + c = 2
< 2 {
If F(X) is continuous,
f (x+c)2 dF(x) J1/2 { f F(x)2 dF(x) } 1/2
J
. F(x)
'Ii (F):: min
o
f
2
= 1/3, and the bound
dF(x)
2.3- 1/ 2 (1 +
is attained with a continuous cdf in
£'
0
2 )1/2 -
0 }
as shown by hartley and David.
Now let F(x) be a step function with at most m steps which takes
on tho valuos
4
0
=
Po < PI < ••• < P
f F(x)2
-
-
-
m-
I < P
-
m
m
dF(x) =
Z
j=l
( P. 1 + P.)
J-
J
~
2
1.
Then
(P. - P . -1 )
J
J
,
and this can be written
p.3
J
,
PJ' = p.J - P.J- 1
•
- 15 The conditions
....1 ,
~Pj
p. > 0
J -
1-F(x)
2
imply
1
dF(x):5
Hence
•
...
3
1
12m2
,
and the Hartley-David bound cannot be arbitrarily closely approached with
a step function in
~
having a bounded number of steps.
Combining Theorems 2.1 and 2.? we can state that if the conditions
of both theorems are satisfied, then
'lieF) =
sup
1" e D
sup
1" e =s
D k+ S
=
HF)
0
In particul;:.u' the conditions of Theorem 2.2 are fulfilled if
'li (F)
=
PI" { f(X, y)
PI" { ••• }
~
or
c }
PI"
f , f(x,Y)l
;: c}
,etc., where
is the probability of the event in braces when X and Yare
independent with the common cdf F,and f(x,y) has the property stated in tho
theorem.
Referring to the remarks following the statement of Theorem 2.1,
we obtain
Theorem 2.3.
~et
conditions
f
Q be the class of cdfs I"(x) which satisfy the
-
m.
x ~ dF(X)
= c.
~
,
i :::; 1, ••• , k
,
- 16 -
F(x)
~nth
given
=
integers~,
we may have A = -
00
0 if x < A,
••• ,
~
F(x)
=1
if x > B ,
and_giveE numbers c1 , ••• , ck ' A, B, whore
and/or B =
00.
~ f(x,y) be a stric~.~.!.~easing
function in each of its arguments when the ether argument is fixed.
PF { f(X,Y)
sup
FeD
=
~
= sup
cJ
Then
PF { f( X, Y) Z c }
F e ~2k+2
The least uppor bound of p(X + Y ~ t) '-Then X and Yare independent,
3.
identically distributed and nonnegative with given mean.
As an application
of the results of section 2 we shall prove the following theoIBm.
Theorom 3.1.
com.mon cdf
F(x)
=0
Let X ~d Y be two independent random variables with
Lot Q be the class of cdfs F ~
F(x).
for x < 0, ~ !J. > O.
sup
FeD
x dF(x) = !J. ~
Then
<2 ,
PF(X + Y ~ c!J.) = 1
if
c
4
if
2 < c
=
(3.1)
f
=
=
--:~
c
1
-c2 - --r
c
5
if 2~c
<.2 ,
- 2
•
- 17 The three bounds are attained with the respective distributions
p(X ::: lJ.}
P(x = 0)
p(X
= 0)
- -2c
,
p(X =
1
1--
,
pCX
=1
:::
=1
C
,.
~ IJ.) -
= cp. )
:::
-c2 ,.
-1
C
Theorem 3.1 should be compared with the solution by Birnbaum,
Raymond and Zuckerman
L- 1 _7
of the analogous problem without the re-
striction that X and Y be identically distributed.
Let M(t, A, IJ.) be the
least upper bound of P(X + Y ~ t) when X and Yare independent, nonnegative, and have the respective means A ~ IJ..
1.2
_7
we have
M(t, A, IJ.)
=
1
::: ...l::t-A
According to /-1, Theorem
- 18 Obviously
sup
FeD
PF(X
+
Y > c~)
=M(c~, ~, ~) = M(c, 1, 1) ,
and we have
M(c, 1, 1)
=
1
==
c=I
:::0
C -
if c < 2
1
2
1
2
if
2<C<%(J+"g)
if
~
c
(J +
/s) ~
c
•
Hence the bound (3.1) is smaller than the Birnbaum-Raymond-Zuckorman bound
if and only if 2 < c < (3
+;5: ) /
Proof of Thoorem 3.1.
2 •
We may and shall assume that
~
= 1. By
Theorem 2.3 we noed consider only edfs FinD which are step functions with
=
m~
4 steps.
Then F is of tho form
F(x)
= P.
J
i f a.J -< x < a.~l
'
J ...
j
= 0,
1, ••• , m,
where
o = a0-<
a < a < ••• < a < a 1
m
m+
2
l
= 00
,
- 19 -
o
< p
=
m
~lma.(P.-p. 1) =
J
J
J-
1
1
,
•
INo have
K(x,y)
Henco the num~ors K..
lJ
K..
lJ
=1
if
x+y.::c
,
=0
if
x + y <
•
C
= K(a.,
a.) satisfy tho conditions
1
J
= 0 or
,
1 ,
if
The sO'llonco (K , K ,
ll
22
followed by a sequonce of ones.
i < if
... , Kmm ) cow'ists of a so'll (mce of zeros
The reasoninG used in tho proof of
Theorem ?.2 shows that any distribution for which thero arc more than
two consecutive zeros or more than two consecutive ones in this sequence
can be replaced by a distribution with less than m stops 1\Thich clocs not
- 20 -
docrease the value of
'!r (F).
Hence for m = 4 we need consider only matrices
l\
K
ij
\\ of tho four
types
000 •
o0
0 0
000 •
o0
1 1
o 011
o1
o1
•• 1 1
1 1
o 001
o0 1 1
o1 1 1
1 1 1 1
1 1
111 1
111 1
o0
1 1
o0
1 1
whero the numbers represented by dots need not be specified.
The corresJ.. . onding matrices
I
IlL..J.J II.
are
II
00.
0 1
0
0
01.
1 -1
0
1 -1
0
0
0
..0
0
IV
III
-1
1 -1
0
0
0
0
o -1
0
0
0
0
0
Ho shall apply L:}nnna 2.1 to show that in evory case there exists a
cdf in D with at most three steps which does not decrease the value of
=
'!reF).
It is sufficient to find integors u, v (1 ~ u < v ~ 3) such that
the equation
(au - a u+ 1) x + (av - a v+ l)v = 0
0
- 21 implies
L x
uu
2
2
+ 2Luvxy + LvY
> 0
v-
•
In case I inoquality (3.4) is satisfied with u
cases II and IV with u
= 1, v
=
3.
In Case III lot u
= 1,
v
= 2,
and in
= 1, v
= 3~
Then
tho loft hand side of (3,,4) is -2xy, and this is nonnegative by (3.3) since
a j -a j +l < O.
Hence we may confine our attention to step functions in D with
=
m~
3 steps.
If m = 3, we have to consider the matrices II K•. /1 of the forms
~J
000
oaa
000
001
aa
aa
001
all
001
011
1
000
all
011
111
111
all
111
111
E
F
1
The corrosponding matrices
A
o0
o1
1
1 -1
ij
0
0
o -1
are
D
C
B
0
II L II
a
oa
1
In applying Lennna 2.1 we have to take u
1 -1
= 1,
0
0
-1
0
0
v
=2
and to show that
(3.3) implies (3.4). This is true for the matrices A, D and E.
cases B, C and F Lemaa 2.1 is not applicablc o
-1
In the
- 22 ~(F)
In C2se C,
2
= 1 - P20
If G(x) is defined by (2.7) with
m = 3, we have
whore t and D2 satisfy the conditions
,
(3.5)
•
Lot D
2
= -1. Then Dl is given by (3.5). Let
which satisfies 0.6).
In case F,
1jr
(F)
t be tho largest number
Then t > 0, G is in ~2' and \If (G) -
= 1 -
1jr
(F) ~ 0 •
pi, and a similar reasoning shows that
this case also can be reduced to a step function with at most two steps.
The only remaining case with m = 3 is caso B.
Hore wo can write
,
where (admitting tho possibility· that F has loss than throe steps)
- 23 -
(3.8)
,
al ~ a <
(3.10)
3
Expressing
C,
c~ a
2 2 2 < c,
2
'!r (F) in terms of aI' a , a , PI'
2
3
'!reF)
= (1
- PI)
+
WG
a •
3
get
2
where
If a 2, a PI are held fixed,
3
W(F)
is a decrQasing function of a •
l
Hence 1,y e mmd.rr.,iz,6
'!reF) by choosing the loast possible value for a •
l
This is the groatost of the bounds given by the inequalities p. > 0 and
J -
a
l
~
o.
If this bound is given by one of the equations P
a distribution in
Hith a
l
=
g20
j
= 0,
WG
got
Hence we may assume that the least value is a
0,
,
l
= O.
- 24 and
(F) is a decreasing function of a 2 wI:eR a and PI are fixed. Tho
3
only lower bound for a 2 which does not necessarily correspond to a distri'tV
bution in
~2
= c - a3 • In this case
is a 2
(l-p )a -1
1 3
2a -0
3
which is a monotonic function of a
fixed.
,
3
(possibly a constant) when PI is held
Hence tho maximum is attained at one of tho endpoints of the range
sup
1Jr(F)
=1
FeD
if
c < 2
•
=
Henceforth we
ass~~e
that c > 2 •
A distribution F in
probabilities
g2
assigns to the points aI' a 2 the respoctive
- 25 -
,
,
whGre
O<a <l<a
- 1- 2
If c
then
~
'Ijr(F)
2u , we have c
1
~
•
2, a case a1roady disposed of.
If c > 2a ... ,
'-
= 0, a case which may be disregarded.
1,.,c are left with the
tID cases
•
In case
(i),
'Ijr (F)
a decroasing function of a 10
If a 1
=
°
~
,
The lower b01.mdfor a
1
is max(O, c-a ) .
2
c-a , than
2
p
1
= 1
,
so that 'Ijr (F) is a decroasing function of a 2.
The 10wi.3r bound for a
2
is
- 26 -
max (1, c) ::: c, and
"lnT;';
0 bta in
2
'¥ (F) = 1 .. (1
1
::: C - - 2
c
If 3 1 = c -
3
2
~
0, than
c-2
so that
¥(F) is 0n increasing function of
th,) some maxirnu.rn of
1\1 (F)
,
3
2
•
Since at) < c, we obtain
<: -
in th8 prl3vious case.
8S
In case (ii),
\jr(F)
a decreasing function of a •
2
Ronce
lolO
lot a
= c/2.
2
Thon
decreasing function of !.l1' and hence is maximized for a
1\1 (F)
1
= 0.
4
= ~.
c
Hence
(J.12)
sup
FeD
=
1\!(F) ::: max
~
3_
lC
1
4
2
'
"2
c
c
~
)
if
c > 2
1\1
(F) is a
Wo get
- 27 -
ThGorom 3.1 now fo11o'Vls from (J.11), (3.12) and tho statoc conditions
lIDder which the bOlIDds are attained.
4.
and lot
l
n
Bounds for P(X + ••• + X >
1
nill
n
(
c).
Lot
1n
t ) donoto tho lea st upper bound of P (X
=
n- 1 (X
1
+ ••• +
X ),
n
> tj.L) ...hon Xl' ••• ,
n-
are independent, identically distributed and nonnegative with mean j.L.
C~TGY'Y
It is oasily soon that for
(J)
ill
(t)
=1
(t) <
-
ill
n
sn
n
if t < 1 ,
s
(t),
G =
1,
2~ .0'
By Markov's inoqua1ity
(1)
(t)
1
1
= t
if 1
=: t~
By Th::;orem 3.1
,
- 28 -
~~( t) bo tho l8'"1st uppor bound of l(X > tlJ.) when Xl'
n
nXn rtrJ indc~0ndent
and nOl1nG b ativo with cownon moan IJ.. Clearly
~
Let
(j)
... ,
f
~ (t) <
n
(j)
-
n*(t) •
It Cbn be verifiod (cf. Jlso L-l, Corollary 2.2_7 ) that
(j) : (
t) :5 M (nt, m, n-m),
wh3ro N(t, A, IJ.) is d:fincd by 0.2).
= 2m
~
n,
In particular, with n
if
and with n
2m
3+
~ ~ t,
4
= 2m,
n eVGn,
+ 1,
if
3n+l+ J5n2+6n+5
< t ,
4n
which holds for n odd and, a fortiori, for n ovon.
On the othor band, for any random v::,riJblos X.1 which satisfy our
P(ln->
assumptions,
(t). In pJrticular, if
n
nt > 1 and X. = 0 or nt with rospective proba")iliti0s 1 - (nt)-l and (ntf\
1
we get
tIJ.) is a low0r bound for
(j)
- 29 -
(l)
l)n
(
( )
t
n t ->1- I - n-
1 n-l 1
v.::n t 2 .
>'%'-~
Honce we h,ve for all positive integors n
(1~.1)
<
t ,
whoro
1
<
Q < 1
n
J
and
1
if
~
whero
o<
2
n
9' < 1 - -
Equation U+.l) is also true for
•
(,l)
~(t), and (4.2) holds for
-30(j) ~~ ( t) if (J +
n
.0) /4
< t.
-
Thus for largo values of t tho known bounds for (j) (t) and
n
."-
(j)"(t) cannot bo substantially improved.
n
-31~1_7
z. W. Birnbaum, J. Raymond,
and H. S. Zuckerman,
/-2 7 H. O. Hartloy and H. A.
-
-
David
,~ generalization of Tshobyshov's inoquality to two dimonsions ll , Annals of
MathEmatical Statistics, Vol. U! (19Lf7),
pp. 76::~
IIUnivorsa1 bounds for moan range and
extremo obsorvation", Annals of lIath •
.§tat., • ~ •
"The extrema of the expected value of a
function of independent r::ndom variables".?
...
"Limits of the ratio of mean ran ge to
standard deviation", Biometri~, Vol. 34
(1947), pp. 120-122.
IlSome remar':s on the inequality of
Tchebycheff U, Courant Anniversary Volmne,
1948, pp. 345-350.
/-6 7 J. A. Shohat and J. D.
- - Tamarkin,
The Problem of Moments, American 11athematical Society, 1943.
"Limits of a distribution function
determined by absolute moments and
inequalities satisfied by absolute
moments", Trans. Amer. Hath. Soc.,
Vol. 46 (1939), pp. 280-306.