NATIONAL BUREAU OF STANDARDS REPORT

1993

TABLES OF E(l/X) FOR POSITIVE BERNOULLI

AND POISSON VARIABLES

by

Edwin Grab

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director

U. S. DEPARTMENT OF COMMERCE

Charles Sawyer, Secretary

cover of this report.

Electricity. Resistance Measurements. Inductance and Capacitance. Electrical Instruments. Magnetic Measurements. Applied Electricity. Electrochemistry.

Optics and Metrology. Photometry and Colorimetry. Optical Instruments. Photographic

Technology. Length. Gage.

Heat and Power. Temperature Measurements. Thermodynamics. Cryogenics. Engines and

Lubrication. Engine Fuels. Cryogenic Engineering.

Atomic and Radiation Physics. Spectroscopy. Radiometry. Mass Spectrometry. Solid State

Physics. Electron Physics. Atomic Physics. Neutron Measurements. Infrared Spectroscopy.

Nuclear Physics. Radioactivity. X-Rays. Betatron. Nucleonic Instrumentation. Radiological Equipment. Atomic Energy Commission Instruments Branch.

Chemistry. Organic Coatings. Surface Chemistry. Organic Chemistry. Analytical Chemistry.

Inorganic Chemistry. Electrodeposition. Gas Chemistry. Physical Chemistry. Thermochemistry. Spectrochemistry. Pure Substances.

Mechanics. Sound. Mechanical Instruments. Aerodynamics. Engineering Mechanics. Hydraulics. Mass. Capacity, Density, and Fluid Meters. Organic and Fibrous Materials. Rubber. Textiles. Paper. Leather. Testing and Specifications. Polymer Structure. Organic Plastics. Dental Research.

Metallurgy. Thermal Metallurgy. Chemical Metallurgy. Mechanical Metallurgy. Corrosion.

Mineral Products. Porcelain and Pottery. Glass. Refractories. Enameled Metals. Concreting Materials. Constitution and Microstructure. Chemistry of Mineral Products.

Building Technology. Structural Engineering. Fire Protection. Heating and Air Conditioning. Floor, Roof, and Wall Coverings. Codes and Specifications.

Applied Mathematics. Numerical Analysis. Computation. Statistical Engineering. Machine

Development.

Electronics. Engineering Electronics. Electron Tubes. Electronic Computers. Electronic

Instrumentation.

Radio Propagation. Upper Atmosphere Research. Ionospheric Research. Regular Propagation Services. Frequency Utilization Research. Tropospheric Propagation Research. High

Frequency Standards. Microwave Standards.

Ordnance Development. These three divisions are engaged in a broad program of research

Electromechanical Ordnance, and development in advanced ordnance. Activities include

Electronic Ordnance. basic and applied research, engineering, pilot production, field testing, and evaluation of a wide variety of ordnance materiel. Special skills and facilities of other

NBS divisions also contribute to this program. The activity is sponsored by the Department of

Defense.

Missile Development. Missile research and development: engineering, dynamics, intelligence,

instrumentation, evaluation. Combustion in jet engines. These activities are sponsored by the

Department of Defense.

@ Office of Basic Instrumentation • Office of Weights and Measures.
NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

1103-11-1107 1U October 1952 1993

TABLES OP E(l/X) POSITIVE BERNOULLI

AND POISSON VARIABLES

fey

Edwin Grab

Statistical Engineering Laboratory

The publication, reprl

unless permission isob

25> D, C, Such permit

cal ly prepared If that

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

part, Is prohibited

dards, Washington

rt has been speclfiirt for Its own use,

FOREWORD

These tables were prepared as part of a continuing

program of research on mathematical statistics and its

applications carried out1 at the National Bureau of Standards

under the general supervision of Dr. Churchill ^isenhart.

Chief of the Statistical Engineering Laboratory® The

Statistical Engineering Laboratory is Section 11.3 of the

National Applied Mathematics Laboratories (Division 11 p National Bureau of Standards), and is concerned with the

development and application of modern statistical methods

in the physioal sciences and engineering.

Jo H. Curtiss

Chief, National Applied

Mathematics Laboratories

A. V. Astin

Director

National Bureau of Standards

TABLES OF E(l/X) FOR POSITIVE BERNOULLI

AND POISSON VARIABLES

by

Edwin Grab

INTRODUCTION

The random variable X is said to have a positive

Bernoulli distribution [l] if the probability that X=x

is equal to (*J)p

X

(l~p) n ”X

/C 1- (l=p) n] for x=l,2,...,n and

0<p<l«, Similarly the variable X is said to have a positive

Poisson distribution if the probability that X=x is equal to

e»m m

x

/x j (i~ e

“ m

) for x=l,2,„.<>, and m>0 o This report tabulates

the functions:

E(l/X|n,p) was tabulated for the following values of the

parameters

:

n

oo

x-1

n = 2(1)20(5)30, p = .01, o 05 (o 05 )o 95 . o99?

n = 21(1)24, p = .01, »05(o05)c50?

n = 26(1)29, p = .01, c05(o05)o45,°

n = 35 and P = -01, „05 ( .05) o35<»

~ 2 ~

E(l/Xjm) was tabulated for thes<= values?

m - .01(. 01). 20(. 10)1(1)10(5)20.

All tables are given to five decimals 0

The need for tables of the above functions arises in many

nr. ’ i 0 f sampling when zero is an inadmissible value of the

variable LlJ, [2]„

COMPUTATION METHODS AND USE OF TABLES

The computation of E(l/X)'1 for positive Bernoulli variables

is a laborious task on a hand calculator* For the ranges of

parameters covered by TABLE I (n is small), there is no

simple approximation of E(l/X)» Stephan [1] presents a factorial

series as an approximation of E(l/X)„ Finkner [2], from Monte

carlo experimentation, suggests l/(np=l) as an overestimate of

me rune ti on with l/np the lower bound* We used as an estimate

of E (l/X)

(3) 1/ (np=q)

(where q = l~p)

»

Included with the tables are graphs of (i=l,2,3) for

p equal to <>5>0 and » 90 « D

1 = E (l/X ) - l/np

D

2 - E(l/X) - 1/Cnp~q)

D

3 = E ( l/X ) - 1/ (np-1

)

Table II gives the relative error jR* = D i/E(1/X)] when n=15

and 30 for the various values of p®

•jV

“

' E(1/Xj will for convenience be used to denote E(l/Xjn,p)

or E(l/Xjm) when there is no chance for confusion*

=* 3 = Linear interpolation within TABLE I will in most cases

produce two significant figures 9 while equation (3 ) 9 for the

probabilities indicated by a footnote to the tables^, is a better

approximation tl&i 1/np or l/(np=l) and produces accuracies of

two or three decimals if not two significant figures 0 As the

magnitude of n increases , l/np=q) rapidly approaches E(l/X),

E(l/X|n p p) was computed by summing the probabilities of

the xth term of the binomial series [ 3 ] divided by x 9 with the

resulting summation divided by l~(l~p) n <>

Two methods were used in calculating E(l/X|m.) 0 Poisson

tables in Fry [43 were used for m = ,1 ( ol)l(l)10(5>)20, The

calculation of E(l/X(m) using Poisson tables is done in like

manner to the calculation of the Bernoulli reciprocal using the

binomial! tables. An alternate method , used in the parameter

range of ,01

(

o 0l) ,20 , is

(I4.) E ( l/X | m) = [Ei(m) = ^loge m]e°m

/(l“@”m

)

[£3p [6]j, [ 7 ]. Values of E(l/X) for m~,l and ,2 provided checks

as to similarity of the methods. The more inclusive tables of

the Poisson distribution by Molina [8] and Kitagawa [9^ could

have been used land are easier to work with then formula (4) ] for very small m values. The results are given in TABLE III,

Linear interpolation within the table will generally

produce two decimal place accuracy. It is suggested for the

range 10<m<40 that l/(m=l) be used for the approximation and

l/m be used for values of m>40 e

[ 1 ]

- u = REFERENCES

F 0 F, Stephan* ’’The expected value and variance of the

reciprocal and other negative powers of a positive

Bernoulli an variate 9 ’* Ann° Matho Stat 0 * 16* 50=61*

(1945) • • — “ — “

[2 3 A 0 L, Finkner* ’’Further investigation on the theory and

application of sampling for scarcity items”* Ins ti tute

of Statistics * University of North Carolina* Mime© 0 Series 30»

[33 National Bureau of

Pi s tribution*

Standards* Table

Applied

Binomi al

[ 4 ] To 0 o Fry* Probability and It 8 s Engineering Uses * D e van

Nostrand Company* Inc,* 45^=4&2* (1956) o'

[51 National Bureau„of Standards* Tables of the Exponential

Function exs> Applied Mathematics Series Tf£T“

[6]

National Bureau of Standards* Tables of Sine* Cosine* and

Exponential Integrals * 1* MT 5? Uo 5, Government Printing

Office* Washington* U 0 C" 0

[ 7 ] National Bureau of Standards* Tables of Natural Logarithms

MT 10* Uo Government Printing Office* Washington*""

[ 8 ] Co Molina* Poisson 8 s Exponential Binomial Limit * D» van

Nostrand Company* Inc « , 1194-7) »

[93 T c Kitagawa* Tables of Poisson Distribution * Baifukan*

Tokyo * Japan* (1955)° ~
*
- 5

-

TABLE I

E(l/X|n,p) £-ir —

1

(")px

(i-P ) n *xAti-(i-p)n

] X

2 3 4 5 6

oOl .99749 .99498 .99247 '. W?7

' .96747

.05 o98718 .97444 ,96178 .94920 .93671

.10 .97368 .94772 .92214 o 89696 .87220

•15 .95946 .91983 088117 .84357 o 80708

.20 .94444 o89071 083898 07894O .74210

• 25 .92857 .84479 .79571 . 73489 .67806

»30 •9H76 o82877 •75158 ,68055 •61583

• 35 .89394 .79594 ,70683 ,62697 .55629

0I4.O 087500 .75744 066176 .57474 .50026

•45 ,85484 •72672 ,61676 •52451 ,44843

.50 .83333 •69o48 ,57222 •47688 ,40132

.55 ,81034 .65330 .52862 .43241 .35890

06Q .78571 o6l538 •48645 .39156 .32231

.65 .75926 .57697 •44622 .35465 •29037

o 70 •73077 .53869 ,40843 •32183 .26305

• 75 .70375 .49543 .37353 .29311 •23989

08O ,66666 ,46237 ,34188 .26829 ,22031

0 85 .63043 •42608 •31373 .24704 •20372*

o 90 •59091 .39189 ,28915

' .22891*"- .18956

• 95 •54762 •36065 ,26803"* .21340 .17734

o99 ,50990 .33843 .25338 .20253 . 16869

iT 1/ (np~q) produces accuracies of two or three decimals 9 or two significant figures; in predicting E(l/x) at this

point and improves as p increases 0

- 6 -

TABLE I (Continued)

n

n

E ( l/X |

n

s p ) = / ,

^ x

^p

X

(l”P)

n A/x[I° (l-p)

n

] x-1

\ n

\ 7 8 9 10 11 j 01

1 .98597

"

' .98257 797998“ 797759

' ^97591

05 .92431 .91200 .89979 .88767 . 87565

,10 .84786 .82400 .80060 .77768 .75526

1

6

.77176 .73763 .70575 .67312 .64277

20 .69715 065461 .61450 .57682 .54152

.25 .62529 .57657 .53185 •59095 .55371

30 .55736 .50592 .55819 .41674 .38010

35 .59551 .44067 .39529 .35550 .32018

40 .53725 .38436 .34016 .30327 .27243

45 .38637 .33604 .29523 .26204 .23567

50 .35195 .29530 .25857 .22911 .20541

55 .30378 .26048 .22864 .20285 .18217

60 .27157 .23358 .20447 .18177 .16361

65 o24436 .21049 . 18477 .16467 .14854

70 .22173 .19151 . 1685 6 .15057 .13608*

75 .20282 .17570 .15505 .13876* .12560

80 .18692 .16238* .15359* .12872 .11665

85 .17341* .15101 .13376 .12006 .10892

90 .16181 .14118 .12523 .11252 .10216

95 .15172 0 13259 .11774 0 10589 .09620

99 .14454 .12645 .11238 .10112 .09192

- 7

-

TABLE I (Continued)

E(l/X|n*p) = n

)p

x

(l-p) n “x/x[l-(l~p) n

] x—

1

12 13 14 15 16

oOl .97253 ,97004 .96757 .96509 "T952F2

o05 .86373 ,85191 o84020 ,82859 081709

olO .73334 ,71193 o69105 ,67069 065087

.15 o61370 .585vi .55938 .53412 ,51007

,20 .50856 .47785 .44932 .42284 .39832

«25 041990 ,38930 ,36163 .33667 .31415

o30 o34780 <>31937 .29433 .27228 .25282

0 35 o2908l .26557 ,24382 .22502 .20868

.40 .24655 ,22471 ,20619 .19035 ,17668

,45 o21243 ol9372 .17795 .16453 .15299

.50 .18601 ,16992 ,15638 0 14486 .13493

»55 .16530 .15131 ,13952 .12945* . 12075*”*

O 6o .14878 0 13643 ,12600* oll707 ol0933

065 ,13532* ,12429* .11493 .10689 0 09991

o70 0 12416 ,11 41

7

,10568 .09837 .09201

*75 .11473 ,10561 .09783 .09112 .08528

080 0 10666 ,09826 .09108 . 08488 .07948

085 0 09967 ,09187 .08521 .07945 .07442

»90 .09355 , 08628 .08006 ,07487 .06997

<>95 ,08814 o08133: .07550 .07044 .06602

»99 0O8425 ,07777 o07221 .06739 o06317

- 8 -

E (1/X | n,p)

TABLE I (Continued

)

n

= ^

( x

)p

x

(l=p)

n ^x[l-(l-p) n

] x=l

pX\ _

.

17 18 19 20

.01' o960l5 .95769 .95523 .95277

.05 .80570 .79443 .78326 .77222

,10 0631^8 .61282 .59460 .57968

.15 048723 0 46556 .44502 .42557

.20 .37565 .35480 .33535 •31750

0 25 .29384 .27552 .25898 .24403

»30 0 23562 .22038 .20682 .19472

»35 0 19442 .18193 .17085 .16104

.40 0 16482 .15444 .14529 ol3717

®45 0 14296 . 13419 ,12643 .11954-

»5o ol2629 .11870* .11198* .10599

®55 .11316* .10648 .10055 .09525

060 ol0255 .09658 .09126 .08650

s>65 o09379 .08841 .08357 .07925

o70 .08643 .08148 .07708 .07312

®75 0 08014 .07559 .07153 .06788

080 .07472 .07050 .06673 .06335

0 85 .06999 .06605 .06254 .05938

.90 .06582 .06214 .05885 .05588

®95 .06212 . 05866 .05556 .05278

®99 .05946 .05615 .05319 .05053
.
TABLE I (Continued)

E (l/Xj n,p

)

n

2

X=1

(*)pX

(l~p)

n”X/x[l-(l~p) n

]

p

V\

21 22 23 24j

oOl .95031 *94786 .91*51*1 .91*297

*05 *76128 .7501*7 .73977 *72920

.10 .55975 .51*312 *52700 *51140

*15 *40718 o38980 .37338 *35788

o 20 *30103 *28287 *27180 *25885

o25 .2301*9 *21822 *20079 *19689

*30 *18389 *17416 ol6537 .1571*2

o35 *15229 * 14444 d3736 .13095

*40 0 12992 *12341 *11752 *11218-

«45 .11336-::- .10780* *10277* *09819

*50 *10061 . 09576 *09136 .08735

V

p \ .

.

26 27 28 29

.0$

,10

.15

,20

>25

,30

.35

,40

.45

„ 70841

o48l?l

o32947

*23582

ol7917

O 14359

oll979*: *10285

*09016

,69820

, 46?60

.31647

,22558

,17140

.13755

,11490-::-

,09874

,08663

,68811

>45397

,30421

,21609

,16425

.13199

,11040*

,09496

,08336

,67814

,44081

,29264

,20729

,15766

,12688

, 10624 - ,09148

,08033

- 10

TABLE I (Continued)

n

E(l/X| n ? p) = npx

(l“P) n"xA[Kl-p) n

] x=l

er 25 30 35 4°

".95053"' .9283" :W53T

.05 0 71871.4. .66830 .62098 .57680

olO O1+9630 .42811 .37110 .32381

<,15 .35326 .28174 .23584 .20130

.20 024688 ol9911 .16588 .14190

.25 0 18765' .15157 .12703 .10937*“*

»30 .15019 .12217 0 10300*“* .08909

.35 .12512 .10239*“* .08671 .07523

.40 0 10731 * .08821

-45 0 09400 .07752

o50 0O8367 .06915

e>55 .07551 .06243

06O 0 06864 .05690

o65 .06299 .05227

c?0 o05820 .04835

«75 .05410 .05597

080 .05053 .04203

085 o0474l .03946

o90 .04465 .03718

o95 .04220 .03515

o99 0 04042 .03368

o 8

o o o

O ' o o o u\ o v\ o v\

CM CM 1—

1

iH o o O o o o

12
«
TABLE II

R VERSUS p, n = 15, 30

n

' n = 30

p

R R R R R R

1 2 3 , 1

0^

OJ .01 -5.90782 2.23351) 2.21903 =2.59052 2.56110 2.53879

.05 -0.60916 7 .03435 5.82748 .00244 =1.72060 =1.9926?

.10 .00599 =1.48501 =1.98200 .22139 =0.13567 »0d6792

.15 .18662 -0.33732 -0.49779 .21126 .02758 =0.01409

.20 .21169 =0.07499 -0ol8248 .16293 .03415 “0.00447

.20792 .00992 =0.08011 .12034 .02256 -0.01504

.30 .18385 .03349 =0.04932 .09053 .01383 -0.02316

.35 .15350 .03391 =0.01x564 .06983 .00850 =0.02803

ohO .12440 .02711 =0.05070 .05532 .00555 ”0.03061

<>45 0 09348 .01969 -0.05701 .04450 .00387 -0.03199

•5o .07059 .01381 =0.06206 .03586 .00260 -0.03297

.55 .06365 .00958 =0.06551 .02915 .00192 =0.03348

.60 -.05091 .00675 =0.06774 .02355 .00141 ”0.03374

.65 . 04051 0 00477 =0.06923 .01894 .00096 -0.03405

.70 .03121 .00335 =0.07004 .01510 .00083 “0 .03413

.75 0 02458 .00230 =0.07068 .01179 .00067 ”0.03425

.80 0 01 826 .00153 =0.07104 .00857 .00036 =0.03450

.85 .01284 .00101 =0.07124 .00634 .00023 =0.03447

.90 .00804 .00054 =0.07138 .00377 .00014 -0.03443

.95 .00369 .00028 -0.07141 .00171 .00006 -0.03442

.99 .00074 .00005 =0.07138 .00035 .00001 -0.03444

- 14

TABLE III

> E (l/x | m)

oo

= / e

~ m m

XAlx(l-e _m

) X=1

m E (l/X j m) m E (1/X | m)

To!”

_ .99750

1 .30 ,9263'gH

o 02 .99501 .40 .Q0244

.03 .99251 .50 .87889

.04 .99002 .60 .85571

.05 .98751+ .70 .83292

.06 .98505 .80 .81052

0 0? .98257 .90 .78854

.08 .98009 1.0 .76699

.09 .97759 2.0 .57659

olO o975l4 3.0 ,43268

oil .97267 4.0 .32963

.12 .97021 5.0 .25777

.13 .96774 6.0 .20779

olU .96528 7.0 .17249

.15 .96282 8.0 .14689

0 16 .96037 9.0 .12776

.17 .95792 10.0 .11302

.18 .95547 15.0 .07181

.19 .95302 20.0 .08280

.20 .95058

i

THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

The functions of the National Bureau of Standards are set forth in the Act of Congress, March

3, 1901, as amended by Congress in Public Law 619, 1950. These include the development and

maintenance of the national standards of measurement and the provision of means and methods

for making measurements consistent with these standards; the determination of physical constants and properties of materials; the development of methods and instruments for testing materials,

devices, and structures; advisory services to Government Agencies on scientific and technical

problems; invention and development of devices to serve special needs of the Government; and the

development of standard practices, codes, and specifications. The work includes basic and applied

research, development, engineering, instrumentation, testing, evaluation, calibration services, and

various consultation and information services. A major portion of the Bureau’s work is performed

for other Government Agencies, particularly the Department of Defense and the Atomic Energy

Commission. The scope of activities is suggested by the fisting of divisions and sections on the

inside of the front cover.

Reports and Publications

The results of the Bureau’s work take the form of either actual equipment and devices or published papers and reports. Reports are issued to the sponsoring agency of a particular project

or program. Published papers appear either in the Bureau’s own series of publications or in the

journals of professional and scientific societies. The Bureau itself publishes three monthly periodicals, available from the Government Printing Office: The Journal of Research, which presents

complete papers reporting technical investigations; the Technical News Bulletin, which presents

summary and preliminary reports on work in progress; and Basic Radio Propagation Predictions,

which provides data for determining the best frequencies to use for radio communications throughout

the world. There are also five series of nonperiodical publications: The Applied Mathematics

Series, Circulars, Handbooks, Building Materials and Structures Reports, and Miscellaneous

Publications.

Information on the Bureau’s publications can be found in NBS Circular 460, Publications of

the National Bureau of Standards ($1.00). Information on calibration services and fees can be

found in NBS Circular 483, Testing by the National Bureau of Standards (25 cents). Both are

available from the Government Printing Office. Inquiries regarding the Bureau’s reports and

publications should be addressed to the Office of Scientific Publications, National Bureau of Standards, Washington 25, D. C.