aPia- *- As.sP : r- ^ A

Statistical National

Was

g'.r.eering La oxy ra cf Standards

:gton 23, D. C.

11 1

NATIONAL BUREAU OF STANDARDS REPORT

171*

PROBABILITY INEQUALITIES OF THE

TCHEBYCHEFF TYPE

by

lo Richard Savage

I

<NBp>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT

1103 - 11-1107

NBS REPORT

25 June 19^2 1744

PROBABILITY INEQUALITIES OF THE

TCHEBYCHEFF TYPE

by

I. Richard Savage

Statistical Engineering Laboratory

The publication, reprl

unless permission is ob

25. D, C, Such pern

prepared if that agr

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

rart, Is prohibited

lards, Washington

been specifically

for its own use.
FOREWORD

Inequalities of the Tchebycheff type have been used in

the theory of probability and in statistical inference „ The

number of these inequalities has become quite large, and to

facilitate their use this report attempts to present a collection of the most important „ Emphasis has been placed on

the precise conditions under which the inequalities may be

used, and on stating them clearly » The accompanying bibliography has been made as complete as possible*

This digest was prepared as part of a continuing program

of research on mathematical statistics and its applications

carried out at the National Bureau of Standards under the

general supervision of Dr « Churchill Eisenhart, Chief of the

Statistical Engineering Laboratory* The Statistical Engineering Laboratory is Section 11*3 of the National Applied Mathematics Laboratories (Division 11 , National Bureau of Standards),

and is concerned with the development and application of modern

< statistical methods in the physical sciences and engineering 0

J® H* Curtiss

Chief, National Applied

Mathematics Laboratories

A., V* Astin

Director * National Bureau of Standards
i

TABLE OF CONTENTS

INTRODUCTION . INEQUALITIES .... ...........

I . Markov i

II. Bienayme-Tchebycheff 5

III. Pearson 11

IV. Birnbaum, Raymond and Zuckerman ... 13

V. Berge 15

VI . Guttman 17

VII. Kolmogorov ............. 19

VIII. Uspensky 21

IX. Wald-von Mises ........... 23

X. Gauss (Camp-Meidell) 25

XI. Narumi, Gauss (Camp-Meidell) .... 27

XII. Peek 29

XIII. Cantelli 33

XIV o Bernstein 35

TABLES .....................

I. Probability associated with deviation X. 43

II. Deviation associated with probability P. 45

EXAMPLES .

ii

1

41

42

BIBLIOGRAPHY 49

PROBABILITY INEQUALITIES OF THE

TCHEBYGHEFF TYPE

I. Richard Savage

INTRODUCTION

Inequalities of the Tchebycheff type give upper bounds

for the probability of events of certain types e In particular they give estimates for deviations from the mean in

terms of the moments.

A selected collection of inequalities of the Tchebycheff

type is given. We have picked them for their diverse nature

and for their usefulness in applied and theoretical work.

In the following there will be first a section where

the various inequalities will be presented with seme notes

on their uses and the conditions under which they may be

used. In several cases more than one form of the inequality

is presented in order to make it easier to work with the

inequality. With each inequality we shall tell what random

variables it is for; that is, whether the random variable is

arbitrary, a sum, or whatever other special conditions are

involved, such as the dimension of the random variables, and

the necessary moments to be used.

Next tables are given that should be .useful for finding

what particular probabilities are associated with a specific

inequality when all the other data is given, and tables that
will tell you what size the variable parameters need to be

to get the desired probability* These tables will be found

useful in choosing which of the inequalities to use* Some

of the inequalities involve several parameters for which

tables have not been prepared*

Several examples are given showing how some of these

inequalities can be used* These examples show various possible useSj, but are by no means exhaustive*

The attached bibliography is complete* References by

Uspensky , Introduction to Mathematical Probability 9 and

Prechetj Recherches theoriques modernes sur la theorie des

pr obabilites , Premiere livre , are recommended as good surveys

of the subject*

It has been found useful to use Tchebycheff inequalities when working with distributions whose functional form

is unknown* In many cases this avoids the assumption that

the random variables are normally distributed* All that is

needed to use these inequalities is good estimates of certain

population moments* Sometimes something is required of the

functional form of the density function of the distributions

that are involved* This is true for inequalities X 9 XI , and

XXIb in the text* However 3 it is easy to verify whether th©

necessary conditions are true for the distributions that on©

is discussing*
In statistical work these inequalities have had several

types of use. In working theoretical problems, it is often

necessary to use these inequalities, for instance, in proving the weak law of large numbers for binomial distributions.

These inequalities are particularly useful for testing hypotheses, and finding confidence intervals for the mean of a

distribution if one has some information about the other

moments. In industrial work, these inequalities have been

used to form ‘’tolerance 1 ’ sets.

Usually one does not have the true value for all of the

parameters that are needed for using these inequalities.

But if one has upper bounds for the parameters, that is for

the moments, then one can use these Inequalities, If one has

run a process many times with the same type of material, then

one usually has a good idea of the variance, even if the

process mean has been shifted, so that in a sense we often

know some of the moments, and in this way we can test for

the other ones.

One must remember that a sample acts as a population;

and therefore, once the moments have been computed for a

sample, all of these inequalities will be true for that

sample; that is, these inequalities will tell you bounds for

the portion of the sample in various parts of its range.

Most of the inequalities presented are for the univariate case. There are several papers that discuss the multivariate case In much more detail; in particular, see Camp,
V

’’Generalization to N Dimensions of Inequalities of the

Tchebycheff Type?” Lesser, ’’Inequalities for Multivariate

Frequency Distributional” and Pearson, ”0n Generalized

Tchebycheff Theorems in the Mathematical Theory of Statistics o” Most of these have been omitted because they are

quite complicated and hard to apply* In particular, for

each inequality that we give we tell for which dimension the

random variables must be; and this is a clue to deciding

which one of the inequalities is applicable to a specific

problem.

Several of the inequalities given make special assumptions on the shape of the distributions that are involved.

All of these special assumptions say that the distribution

has an unique mode. In Narumi 8 s article ”0n Further Inequalities with Possible Applications to Problems on the Theory of

Probability” the opposite case is treated, where the distributions have an unique minimum and increase as you go away

from it. This case did not seem to be as important as the

other, and therefore it was omitted.

In Winsten 8 s article, ’’Inequalities in Terms of Mean

Range,” there are inequalities that involve the ranges for

various sample sizes. These inequalities will undoubtedly

prove useful in the future; but they are not entirely analogous to the Tchebycheff inequalities, and therefore were

(

!

omitted
vi

Inequality I contains many of the other ones as special

cases, which is a little surprising since this is the simplest

of all the inequalities o This results from the fact that I

is true for any positive random variable, X, that has a finite

expected value,, In particular, la is derived from I by replacing X by a sum of random variables, II is obtained by

replacing X by a random variable that is of the form of the

square of the difference between a random variable and its

expected value. One can derive many of the other inequalities in this manner.

In cases where the inequality is given only for the random variable X minus its mean, there are also inequalities

for a sample average minus the expected value of that average.

Most of the inequalities are given for the probability

* of the random variable being greater than or equal to some

number; there are opposite inequalities for the event that

the random variable is less than or equal to the same number.

These are the same expressions, having the inequality reversed

within the probability symbol: the * greater than or equal®

symbol being replaced by a «less than’ symbol, and the righthand side being replaced by one minus the original right-hand

side. Also, one may derive a new set of inequalities from

the ones given here by interchanging - with <, and - with >,

If the variables are continuous this adds nothing new.

As given, some of the inequalities are very weak, for

the right-hand sides may be greater than one; but a probability
vii

is always less tharx or equal to one, so to get the most out

of these inequalities the right-hand sides should be interpreted as the minimum of whatever is there now or one

„

If one considers whether most of these inequalities can

f %

be improved, that is, can the right-hand sides be replaced

by smaller quantities, one will find that this cannot be done.

That is, there always is some distribution that satisfies the

conditions necessary to use the inequality, and such that the

left-hand side equals the right-hand side, Of course this will

only occur for certain exceptional cases; but you might have

that case. If you know that you do not have that case, then

there might be a better inequality that you could use.

In the following, unless otherwise noted, lambda (X)

is any positive number, EX equals the expected value of the

random variable, and will be denoted by yu^ and if needs be

this will be given a subscript. The expectation sign E will

be used to denote other expected values depending on the

argument that follows it. That is, will be the

variance and will be denoted by a2

„ In general, as far as

moments go, we will use the notation of Cramer (I9I4-6) , The

symbol P(A) means the probability of the event A.

(

»
- 1 -

I „ MARKOV

P(1 £ X) s l/X

P(X £ X) S yu/l

Random Variable; X

Restrictions; X is non-negative , that is, P(X<0 = 0

Dimension; One

Moments; n = EX

(i)

(!')

References ;Cramer (19^4-6); Frechet (1937)

Notes

;

1„ This is a fundamental inequality from which

inequalities la, II, Ila, IIb,XII, IV, V, VI, XII, XIII, XV

and XVa may be derived by picking X as a special function of

the random variables that are of interest,

2 0 In using this inequality, note that you need

to know only one moment, or you are testing an hypothesis

about only one moment,

3o By itself this is rather a weak inequality,

for the probability is bounded by l/X ; this is of course

to be expected, since we are only using one moment, and

therefore we have very little knowledge about the distribution that is involved, or at least are only using very

little of this knowledge.

- 3

-

la

,X P(£ *X) * 1/X

n n

P ( 2 Xi/ Z *X) - l//t

i=l i=r

Random Variables n

(la)

(1A» )

X = 2 Xi

i=l

Restrictions? Each Xi is non-negative

Dimension: Each Xi is one -dimens i onal , but actually the

Xi may be considered as one observation on an

n-dimensional random variable , i.e. the X^ may

be dependent

»

Moment s /H = EXi

n

s 2 n±

i=r Z

1

References :Cramer (19ij.6); Frechet (1937)

Notes :

le This inequality is formally the same as I,

but shows how I may be used where the random variable that

is of interest is actually the sum of several random variables

,

2. It is clear how this inequality is derived from

the first one, for it is the same as that one, except that the

random variable may be written in two ways, that is, either

as X or as a sum of Xj_ „
.
II o BIENAYME-TCHEBYCHEFF

(II)

(II’ )

(II !

!

)

Pdx^l zZo) % i/A 2 . P(X - Xo +^u or X-yU-^o)” l/i2

P(|X^| *X) * a2

/. I

2

Random Variable: X

Restrictions: None

Dimension: One

Moments: o 2 = E(X~yu )2

References ;Cramer (1946); Uspensky (1937); Frechet (1937)

Notes

:

L This is the standard Tchebycheff inequality

for one random variable „

2 c V/e now find that the probability is decreasing

as l/i , which means that the probability of large deviations

from the mean becomes quite small 0 It is to be noted that

for the normal distribution and for large X this is actually

a very poor estimate of the probability of large deviations,

for there the probability of a large deviation is smaller

than e to the minus one half , but for intermediate values

this is not a bad approximation.

If one has fairly good estimates of a, then this

inequality may be used for the testing of hypothesis about

the mean, and for finding confidence intervals for the mean c In many industrial applications this inequality is used for

getting an estimate of how much of the production will be

near the mean of the process, where one has a good idea of

the variance

o
.
(II*)

(II* 1 )

- 7

-

Ila.

P(/n |X -

yu| ^Ao) S l/A2

P( |5t -^| 2 A,) - o

2/n(AJ 2

n

Random Variable: X = 2 Xj_ / n

1=1

Restrictions: If i ^ j, then X^ and Xj are uncorrelated;

that is, E(Xi-EXi) (Xj-EXj) = 0

Dimension: One

Moments

:

A

EX*

a

2 = E(Xi- fx)

References :Cramer (19i^6); Uspensky (1937); Frechet (1937)

Notes

:

1» This is one of the most useful of the

Tchebycheff inequalities,, We can use this whenever we have

sample averages of identically and independently distributed

random variables.

2 0 This inequality gives us the square root of

the sample size law. That is, \T - yu| is of the order of

magnitude l/v/n „

3. The uses of this inequality are much like

those of II, but we can also use it In the case where we are

working with sample averages.
c
- 9

-

IIb<

P ( | X - fi\ -A/o) - l/l2

P(|X«yi| * X) - ^

Oij^

Random Variable:

x = Z

i=l

1

Restrictions: None

Dimension: One

Moment s

:

yU;L “ , Z

1 =

Oij = E(Xi-^ii) (Xj-^j)

n n

°

2 = z

, °iJ

i=l J=1

(lib)

(lib'

)

Reference

:

Uspensky (1937)
- 10 -

Notes

s

1, This is another form of the Tchebycheff inequality 0 In this case the random variables forming the sum

in which we are interested may have different variances and

they may be correlated; we still get the same type of inequality o

2„ Actually, if all of the covariances are about

of the same size as the variances, then this inequality is

likely not to give much better bounds than l/X °

3c A little thought will show how to write this

inequality in terms of the sample average rather than the

sample sum„ In the remaining inequalities we will only give

the inequality for the sample of one observation* but the

way to change it to sample sums or sample averages will have

become clear from the various versions that we have given of

these first two inequalities

„

ij-o Another point of interest is the fact that

although the inequalities often require exact estimates of

certain moments before they can be used* it is possible to

get similar inequalities by substituting upper bounds for

the moments that are involved 0 For instance, in this inequality we might not actually know the value of a* but from

previous experience we might know that it cannot under any

circumstances be greater than* say, 6 a In this case, if we

use 5 instead of a, then the inequality will not be as good

as the given one* but still may prove useful « This technique

may with some care be used for all of the later inequalities

that we give, as well as the earlier ones.
III. PEARSON

P(i*--4i

IT77

P(|X-yu|

l/r (hi)

* X') - Pr/a>) r (in')

£ X" ) s Pr (III* *

)

ar O,”) r

Random Variable: X

Restrictions: None

Dimension: One

Moments: ja. - EX

o2 = E(X-^u) 2

Pr = E|X-^u( r

References :Pearson (1919)? Narumi (1923)? Frechet (1937)

Notes :

1. In order to use this inequality we need

an absolute moment of the random variable.

2, If we know several of the moments and wish an

inequality for a particular X

,

then we should pick that moment

that makes the right-hand side of III* the smallest for that

particular X •
a
IV o BIRNBAUM, RAYMOND and ZUCKERMAN

P( Z Z l

2

) -2 l/*2

Z 1/ti2 (IV)

i=l a

i

2 t*2 i=l

n

P( 2

i=l

n

(Xj "yuj_

)

^

ti2 a-)

2

) (A')

(IV' )

Random Variables X ~{X\ ,

, ,,, Xn ) Restrictions? The t^ are arbitrary positive constants

Dimensions n

Moments: = EX^

Oi2 = EfXi-^i ) 2

Reference s Birnbaum, Raymond and Zuckerman (191+7)

Notes

s

1, This inequality gives a geometrical result.

That is, it gives the probability of the deviations from

the various means that are involved to all lie within a

hyperellipse. And of course if the ot's are all equal,

this becomes a hypersphere.

2, From the first comment it is clear that this

sort of inequality could be used for bombing problems, etc,

3, Although none of the covariances occur in

this inequality, it is applicable when the covariances are

not equal to zero. The next inequality will show how the

information about the covariances may be used in order to

get a better result.
•

••

.
«= 13 ”

Vo BERGE

Random Variable: X = (X^,X2 )

Restrictions: None

Dimension: Two

Moments: yui = EXi

ai2 = EUi-^i ) 2

Ql,2 = E ( X1 "y^l) ( V2 "^2^

/> = ^2

A °l a2

References

:

Berge (1938)
- 16 -

Notes

:

lo This inequality tells us the probability of

falling outside of a rectangle about the means for a bivariate sample

o

2 0 This inequality uses the fact that the random

variables may be dependent, and therefore in order to apply

this inequality one needs actually to have some knowledge

about the correlation,,

3 o In this case if we have no knowledge about the

correlation, then the worst that can happen as far as the

value of the right-hand side goes is for the correlation to

be equal to zero, and then we still get a fairly good results

And the best that can happen is for the absolute value of the

correlation to be one, and in that case we get the best inequality of this type„
- 17

VI . GUTTM&N

p [(T-^) 2

- ^

+ q

2 y4{Hfp

- 1 5 V*2 (vi)

•

nni

~ Random Variable: X = —

Restrictions: Xi are identically and independently distributed

Dimension: One

Moments: yti = EXj_

o

2 = EfXi-^i ) 2

and define

n 5

Z (Xi-X) 2

S

2 -±=L,

t

References

:

Guttman (I 9I4

.8 ); Midzuno (19^0)
Notes

:

1, This inequality is applicable whenever Ila can

be used; however, this one takes more computing than that one

does, and that is its only disadvantage. If we compare those

two inequalities, the only way they differ is by the quantity

on the right of the inequality sign within the probability

statement. For this inequality that quantity is a random

variable, and it is not for Ila, It turns out that the expected size of this random variable is much smaller than that

of the quantity that occurs in Ila, and therefore it would

seem to be a much better inequality to use than Ila,
VII. KOLMOGOROV

P (T % yn oA) 5 l//l2 (VII)

i

Random Variable: S-j_ = 2 (Xj-^uj) ; i = l,2,...,n

Let T = max [ | Si| | S-jJ ,... | Sn | ] l<i<n

Restrictions: X^ and Xj independent and identical

Dimension: One

Moments: = EX^

o2 = EfXi-^i ) 2

References :Uspensky (1937); Frlchet (1937); Kolmogoroff (1928)

Notes :

la This inequality can be used whenever Ila is

applicable , although the use of this one is a little

different than that, A typical use of this inequality is

in the extreme-value type of situation. For instance, if

we are putting together an assembly we might ask what is

the probability that the cumulative error ever exceeds a

certain quantity, and this inequality would give the answer.
fli

(II

1 - v ' ' • f

(
VIII. USPENSKY

P(X- u ^ \)

a

^ if X< 0

if 0

Random Variable: X

Restrictions: None

Dimension: One

Moments: p. = EX

o

2

= E (X-^) 2

(VIII)

References :Uspensky ( 1937 ) Notes ;

1. This inequality is applicable whenever inequality

Ila can be used c

2 0 In this case we are interested in one-sided alternatives, that is we are thinking of the case where large

positive deviations from the mean would be the things that we

wish to detect. This occurs, for instance, whenever one is

using one-sided confidence intervals or one-sided test regions.

3. The derivation of this inequality essentially

depends on the Schwartz inequality.
f . ' ' • ' i
23

IX. WALD“VON MISES

P(|X- u | “ t)

P( |X-ya|<t) £ 0

> t? - (3 r

tr

> Pr * tr

“ t' r - tr

Random Variable

:

X

Restrictions? None

Dimension: One

Moments: jx - EX

3r = E|X-/i| r

3 s = Elx-^l®

Let s > r

,

then

|t°i- = g

if t S t° (IX)

if -p o V» -p - to

if t 0 * t

if op

-p VI (IX')

if to S t » t0

if P

VI o

p

it°r = 3,

Pr * tr

_ Ps - ts

t»r- tr t» s » t

s

This defines tg, t^, and t' in terms of p r , p s> and t
References sWald ( 1938 ) | von Mises (1939)

Notes %

lo This sort of inequality is to be used where we

have two absolute moments at our disposal and wish to get

the most out of them 0 The same type of problem also arises

when we have more than two moments, and the answer is given

in Wald (1939) o

2 e These inequalities are harder to use, and it

is doubtful that one will use them in practice e
X. GAUSS ( CAMP -MEIDELIi

)

p ( | x - yu0 1 £ M) 5

9"Xj

Ptlx-^l ifa) S |

* if I s

(X)

(X* )

Random Variable ; X

Restrictions s X has a density function with one mode^iQ

Dimensions One

Moments? ja ~ EX

a

2 = E ( X =*yu ) 2

V 2 = a2 + (fi ”ya0

) 2

1 3 1 =

References sCramer (1946); Narumi (1923)1 Frechet (1937)

Notes

;

1 c As far as having knowledge of the necessary

moments to use this inequality* its application requires the

same knowledge needed for use of inequality II, It is also

necessary to know or have an estimate of the mode; but since

that is usually the same as the mean (always for symmetric

distributions)* this is not much of a handicap,

2 a In practice many distributions have this type

of behavior* decreasing as you go away from the mode* particularly the normal* log normal* x

2 for enough degrees of

freedom* and the t -distribution,

3o If this decreasing property actually is true,

then this is a better inequality to use than II, for it

essentially divides the bound by 4/9

«

- 27 -

XI, NARUMI j GAUSS ( CAMP -ME I DELL)

ifo^xs rb

5+1 (XI)

<j^. if

b

r r+1

- -i- if b

A

r

Pflx-^ol ? ^r) lf *-

(i +r)

r

i-l/r- (XI')

W a) lf

( r+;yT^iTr

Random Variable; X

Restrictions; X has a density function f(x)„ f(x) has an

unique maximum in the interval (yciQ-b

, ^/UQ+b) at /Uq

and b > 0; use (XI) if b is finite, and (XI') if

b is infinite

.

Dimension; One

Moments; (?£)

r = E|Xyi0 | r

References

;

Narumi (1923); Frechet (1937)
Notes s

1. In these inequalities we have used absolute

moments about the modes of distributions , instead of the ordinary procedure of using them about the means

«

2 e This case is likely to occur with the b equal

to infinity, that is the case where there is only one model

however, cases where the interval of a decreasing distribution density function is finite occur sometimes when we are

sure that the distribution behaves in a nice manner for small

values but that there might be little isolated humps of

probability at the extremes 0

3o Actually, in most cases we would probably use

inequality (X)

,

which is a special case of these inequalities

this is why it has been given by itself «,
29 -

XII . PEEK

P(|X-n| ~ to) -

' t 2 -2y+i

Random Variable: X

Restrictions: None

Dimension: One

Moments: ya = EX

a2 = ECX-ya) 2

6 = E | X-m|

/*=" 6/0

(XII)

References :Peek (1933)

Notes :

1 0 This inequality is much like that of II , except

that here one needs to know 6, the mean deviation. If one has

this additional information, this is a better inequality to

use

.
,

(I
Xlla®

P( I X-> Al| ~ to) k

9

A (Xlla)

Random Variable? X

Restrictions? X has a density function whose only mode is

its mean

Dimension? One

Moments? u = EX

o

2 = E(X»/u) 2

6 = E[X«yu|

/° = 6/o

\

References ?Peek ( 1933 ) Notes ?

I, This is an improvement over (X), but can only

be used if one has an estimate of the mean deviation,,

XIII o CANTELLI

p( |x-/i| * \) s Bn

<

U?-pr

')^"+ Vzr P:? Pr

Random Variable: X

Restrictions: None

Dimension: One

Moments: = EX

p2r = E | X -yu [ 2l>

Pr

= E|X-/!| r

If r < P2r

Pr

P2r ” (V

(XIII)

? -r P2r < nr

References gCantelli (1910); Peek (1933); Frechet (1937)

Notes :

1 0 This inequality has probably no great practical

use, but it has several special cases that might be found of

use o

2 0 Of course in this case we need information

about two moments.

- 35 -

XIV, BERNSTEIN

P( |X-yu| U) S 2e

-^2/(2o2 +20^

Random Variable? X = ZXj.

Restrictions; X^ and Xj are independent

Dimension; One

Moments; ^i± - EX^_

, yu =

o±2 = E^-^)2

, a

2 = 2aic

<5 c cm 2 sI C

S5=>2 E(Xj_ -yu^) - •— for all integers s and

some constant Co. (C > 0)

(XIV)

References ;Acceptance Sampling (1950) | Craig (1933); Prechet

(1937); Uspensky (1937)

Notes

;

1, This is a very nice inequality in the sense

that the remainder term goes to aero very rapidly

„

2 0 The one difficulty in applying this inequality

is that essentially one has need of knowing all of the

moments of the distribution that is at hand, or at least one

has to know an upper bound for them; the next inequality

treats the latter case.

XlVa,

P(|X-yu| H) S 2e -^2/( 2 o2 +|m^) (XlVa)

Random Variable:

Restrictions: P(

pendent

Dimension: One

Moments: - EX

X = ZX±

> m) =

, /a =

0 Xi and Xj are indeOi2 = EfXi-yui ) 2

References %Fr^chet (1937); Uspensky (1937)

Notes :

1* This is the case where the distribution is

bounded ; and as a result we have an estimate for all of the

moments, and therefore can use the previous inequality. If

A/ is fairly large, this is one of the best inequalities

given in this paper for this situation.
-
- 39 -

XlVb

P(|X-/i| ^ aX) - e ”^2/2

Random Variable: X - ZX^

Restrictions: X is symmetrical about pi , i„e 0 :

E(X -^) 2i+1 =0 5 X±, Xj are independent

Dimensions One

Moments: pi± = EXi , pa - Zpi±

(XlVb

)

E (Xi-yUl) 21' ^ (^) k (r an integer)

i

2 = E(Xi -

/p1 )

: 2oReferences sUspensky (1937)

Notes s

1, It is not too uncommon to deal with symmetrical

distributions, and therefore from that standpoint this inequality can be used, However, the condition on the moments

is rather stringent. This condition is true for the normal

distribution.

- -

THE TABLES

Essentially Tables I and II are inverses of each other,

column by column,, The columns of the tables are associated

with the Inequalities as follows % Column Inequalities

1 (I), (la), (la*)

2 (II), (II*), (Ila), ( lib) , (VI), (VII)

3 (III with r = 4)

4 (X)

5 (XI* for r = 4)

6 (VIII)

7 (X IV), (XlVa), (XlVb

)

The first table answers the question how large the

probability is associated with a specific '’deviation*1 A, „ The second table gives the ‘’deviation” associated with specif -

ic probabilities P,

An example Is now given showing how to use these tables

„

Suppose we have a sample of nine independent observations from

a distribution whose variance is four units squared,, (1) What

is the probability that the sample average Is more than one

unit larger than the population mean? (2) How far above the

population mean could the ninety percent point of the distribution of sample means be? (3) If we knew the population had

an unique mode at its mean, could these results be improved?

Anss It is clear that for questions (1) and (2) we need inequality VIII, and for (3) we need X 8 „
- 42 -

Let us first evaluate the necessary constants. In

particular X is a sample average and therefore = 4/9 ° Also in X 8 we need s = 0 and o ~ 2/3

To answer (1) we note = 1 and we need to evaluate

cj2 „ 1

02 + Z

2 TTJ7U °

Column (6) Table I for A - 3/2 ~ 1.5 shows ” »3077 and

this is the answer. The answer to (2) Is in Table II, column

(6) £ for P ~ .10 gives an answer of 3 units for 3/o, thus

I s 3 o = 2. The corresponding results using X 88 which melees

use of the unique mode but not the one-sidedness s using

si - 1 and P = o80 we have -= using column ( 4 ) — the answer

.198# and (with interpolation) we have that ill = 1.5# or

o

' - I# respectively.

Thus for this problem it seems better to use inequality

X 8 8 than VIII

These tables will facilitate choosing which inequality

to use when several are available , by comparing the associated

probabilities (deviations) with the deviation (probability)

of interests thus making it possible to choose the inequality

that gives the smallest probability (deviation) for the

problem at hand.

(

EXAMPLES

EXAMPLE ( 1

)

. Assuming that all soldiers are between

sixty and seventy-eight inches tall, what is the probability

that the average height of five hundred soldiers Is more than

on© inch away from the average height of all soldiers?
TABLE I

PROBABILITY ASSOCIATED WITH DEVIATION X,

Probability (1) (2) (3) <U) (5) (6) (7)

associated 'sXswitb 1

X

1 1

33

i* a? if if

1

1+ X

2 e =A2/2

loO 1,000 1,000 IcOOO ,666 .6096 .5000 .607

1.5 .667 .666 . 1975+ .196 .0809 .3077 .325

2oO ,5oo ,250 .0625 .111 ,0256 .2000 .135

2.5 ollOO oi6o ,0256 .071 ,0105= .1379 .066

3c0 .333 .ill .0123 .069 .0050 ,1000 .011

3.5 o286 .082 ,00 6? .036 .0027 .0755= .002

6.0 c250 c062 ,0039 ,028 .0016 .0588 .000

6.5 o222 .069 .0026 .022 .0010 ,0671 /

5o0 .200 „060 ,0016 ,018 .0007 .0385=

5.5 ,182 .033 .0011 ,015» .0005= .0320

6.0 ol6? .028 .0008 ,012 .0003 ,0270

v 6,5 .1514 .026 o0006 ,011 .0002 .0231 .000
;
- 45 “

• TABLE II

DEVIATION ASSOCIATED WITH PROBABILITY P

Deviation

associated \ with

p \

(1)

1

p

(2) (3) (4) (5) (6) (?)

1 1 2

3/P

u

jP7E yp ./

2 l0Se ;

c99 1,0101 i,oo5o 1.0025 .6700 .802 . 1005= .1418

o 95 1,0526 1,0260 1,0129 .6840 .810 ,2293 .3202

o9 1.1111 1.0541 1,0267 ,7027 ,821 ,3333 ,4590

o75 1*3333 1*1547 1.0746 ,7698 .860 ,5773 ,7585

o5o 2. 1.4142 1,1892 ,9428 .951 1 . 1,1774

.25 4, 2. 1.4142 1,3333 1.131 1,7321 1.6651

.10 10. 3,1623 1,7783 2,1082 1,423 3, 2,1460

,o5 20, 4.4721 2,1147 2.9814 1.692 4,3589 2.4477

,01 100. 10, 3 ,1623 6,6667 2,-530 9,9499 3,0348

,001 1000, 31,6228 5,6234 21,0818 4 c 5oo 31c 6070 3,7169

|rH

- 47 -

Solution s Although w© have a finite population, it is safe

to assume that the measurements in our sample are independent o The largest possible variance occurs if half the

soliers have height sixty inches, and half have height seventy-eight inches, in which case the variance is eighty-one

inches squared » First apply inequality IXa„ Here X® s 1

inch, = 81 inches squared, and n ~ 500 o Thus the answer

2 fl-j is = 0 162 0 W© may also apply inequality XIVa c 500

Here X " 500 inches, o* = 500 ° 81 inches, and m = 18 inches

„

The probability is oil, and thus for this example we see

that XIV gives more precise results than II

„

EXAMPLE ( 2

)

o In the course of deposit and withdrawal

transactions, such as money in a bank, or radioactive material

in a hospital, one often wishes to control the absolute error

That is, in a sequence of, say, one hundred transactions (a

day's activity) one does not want one's books to differ from

one 9 s assets, at any time, by more than some fixed amount,

say 1,000 units 0 Let us assume that the variance due to

errors of measuring and of counting for each transaction is

400 units squared (this value being obtained by previous experience) o Th© question then naturally arises 1 what is an

upper bound for the probability of having an accumulated

error of more than 1,000 units at any time during the day?

Solution; Inequality VII is suited for this problem 0 Here

n TM 100, a = 20, and >/n a

X TM 1,000„ Thus X -

= 5, and \/}3

' = 1/25 - »04« If instead of 100 transactions
there had been i|00 9 we would have n = I4.OO ^ a = 20*

/note 1,000, \ = fk

1 „ 00 n 1 nnn

2 « 5 » and the resulting

probability is at most

EXAMPLE ( 3

)

» From previous experience let us assume

that the correlation between two variables (height and weight*

rainfall and crop yield) Is at least 0 8 „ If a sample of

twenty-five is made on this bivariate distribution* what is

an upper bound for the deviations from the population means

that will not be exceeded more than ten percent of the time*

where deviations are measured in standard units?

Solutions Here we can use inequality Vo First we must solve

the following equations

or

Thus P 5 h or hzih a 4) s a ,

or pdf."/1

1 5r k or ly-^Us it) k) sS a±

”y

Thus both sample means are within o 80 standard units of

their respective sample averages with probability
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„

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10, 190-192,

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!l Sur deux theoremes relatifs aux

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Stat-istica, Rljswijk , 3, 201-20O,

J, USPENSKY (1937) ^ Introduction to mathematical probability , McGraw-Hill Book Co,, New York", 373-3801

A. WALD (1938), nA generalization of the inequality of Markoff

Annals of Mathematical Statistics , 9_, 2l+!+-255o

A, WALD (1939)* "Limits of a distribution function determined

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Co Bo WINSTEN (191+6), "Inequalities in terms of mean range,"

Biometrika 9 33 , 283-295.
. ' ci
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