SOME TABLES OF THE NEGATIVE BINOMIAL DISTRIBUTION … · MEMORANDUM RM-4577-PR JUNE 1965 SOME...
Transcript of SOME TABLES OF THE NEGATIVE BINOMIAL DISTRIBUTION … · MEMORANDUM RM-4577-PR JUNE 1965 SOME...
VIEMORANDUM
?.M-4577-PRTUNE 1965
SOME TABLES OFY'• THE NEGATIVE BINOMIAL DISTRIBUTION
AND THEIR USE
Bernice Brown
-- / - -
PREPARED FOR:
UNITED STATES AIR FORCE PROJEM;T RAND
_________Th7 r•i [1 DeCorpo7ateo
5ANTA MONICA CAtIFO'lrA------
biA
MEMORANDUM
RM-4577-PRJUNE 1965
SOME TABLES OFTHE NEGATIVE BINOMIAL DISTRIBUTION
AND THEIR USEBernice Brown
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_________________-- 74e~ 1Q4W I I I )eeAemauw
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In this Memorandum the author presents probability
tables of the negative binomial distribution for some
useful sets of parameter values. This distribution may
be used as a frame of reference for the study of the
demand for replacement parts.
M T~r -
SL,94ARY
This Memorandum presents tables giving the values of
the individual terms of the n,.gaivc 'Ainomial distribution
for 130 pairs of parameter values in Part I. Part 2,
giving the cumulated terms ,)f the same distributions, is
in a form directly usable for solving pjoblems which ze-
quire the determination of probabilities of the occurrence
of not more than x events.
The negativ% binomial .'],..ribution is described and
illustrativ., exat-t.ples are even which use this distribution
as a tool LI- Lhe study of demand for replacement parts.
The reterence- indicat -hat the negative binomial
has been us1 ,d in a wide vazL:,'•y of applications to bio-
logical research.
SOME. TABLES OF THE NEGATIVE BINOMIAL DISTRIBUTION
AND THEIR USE
L, INTRODUCTION
The research worker in the field of logistics "s
frequently required to predict the demand for spare parts
in order that effective decisions can be made in the areas
of procurement, distriuution, and stockage policies. Many
such decisions are made on the basis of a specialized
knowledge about an individual line item.
For routine decisionmaking on a large scale, it is
useful to have a small number of known probability distri-
butions which can be used as a framework to provide the
basic assumptions from which predicto.is of future demands
can be made. If demand data are available, an analyst
may study the sample data and make estimates of the param-
eters of the underlying demand distribution. The informa-
tion in the sample will often be scanty, e.g., six mornths
experience at one base or thrce months of system-wide
experience. A frame of reference is needed to study the
limited data.
Some examples of studies in which it was necessary to
make an assumption about the demand for spare parts in the
population may be found in the study of procurement defer-
ral by Petersen [Il, the design of flyaway kits by Karr,
Geisler, and Brown [2], and by Fort [3], the prediction of
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demands by Goldman [4], and Geisler and Brown [51, the
study of stocka,;e policies by Ferguson [6], Ferguson and
Fisher [7], and Petersen and Geisler [8].
There are many distribution functions which could be
discussed. Only two are presented here, the Poisson and
the negative binomial. These two were chosen because they
are probability distributions of a discrete variable, and
demand for spare parts is by nature integral, and also
because we believe, hopefully, that they may describe the
universe of denmands from which the sample emanates. These
distribut'ons are attractive also because of their easy
computation.
2. THE POISSON DISTRIBUTION
When we deal with phenomena involving events that
occur randomly in time or particles that are randomly
distributed in space, the Poisson process is the model
used. An experiment is performed and "events" are tallied.
These events can be described by a function x = x(t), which
gives the number, x, of events observed during the first
t units of observation fur &ll values of t from 0 through
T (where T is the total number of observations). The re-
sults of such an experiment may be described by the Poisson
distribution if the events occur r in the sense of
the following definition. If any number of events x are
observed in any amount of time t, and if the points of the
occurrence of the x events are indapendently and uniformly
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distributed between 0 and t, then the process may be de-
scribed a3 random. If the probability of the event is
small but a large number of independernt cases are taken,
the number of occurrences is likely to be distributed in
the Poisscn series. This distribution may be thought of
as an approximation to the binomial distribution when the
probability of occurrence of the event is small, that is,
if Np is large relative to p and N is large relative to
Np (where N is the number of trials and p is the probabili-
ty of the occurrence of the event in a single trial).
Let us try to visualize what the situation mig'at be
in regard to demand for spare parts upon the Air Force
Supply System. Suppose a mechanic inspects an actuator
on each of che 18 planes of a squadron, once each month
for two years, and requests a replacement part whenever he
judges that the actuator has failed. Assume that the fail-
ure rate for this part ha8 been found to be 25 in 1000 (i.e.,
1 in 40). We may then think of this situation as being
represented by a binomial distribution (p+q) 3 , where
N a 432 (18 inspections per month for 24 months), p a .025,
q a .975 (i.e., q a 1-V, the probability of nonoccurrence).
But the data available to us Is monthly data by squadron,
and we are interested in determining a squadron demand
rate per month. The monthly demand data for the actuator
looks like this: 0-2-O-0I-O--0-.l-O--l-----2-0---I-l--0-
0-1-0-1. These numbers are ordered in time, the first
observwtion being January, 1956 and the last one December,
1957.
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This is a "natural' for the use of the Poisson approx-
imation to the binomial, since Np is large relative to p
(432 to 1) and N is large relative to Np (40 to 1). If
we assume that the Poisson law describes the data and use
as the parameter of the Poisson the mean demand per squadron
month, i.e., 18 x .025 - .45, we will find a satisfactory
fit with i0 ny.nths of no demand, 7 months of demand for
one part, and 2 months of demand for 2 parts.
The Poisson (like the binomial) is a distribution of
a discrete variable arising from enumeration data using
integral values only. Its basic characteristic is the
uniform probability of the occurrence of the event. In
contrast to both the binomial and the normal distribution,
it is defined by a single parameter, the mean. The variance
is equal to the mean. The Poisson density is represented
by the probability function P(x) a e-" mX/xI., where
x a 0,l,2,3'", and m a mean.
Tables of the Poisson distribution have been widely
published; for exaple see [9), 110), [111.
3. TI NAM TIYK Z UCIAAL MDZTBU'ION
But the aircraft demand data described in Sec. 2 does
not present this picture for many of the parts, and the
Poisson distribution often has not been a good fit for the
series of observations. In many cases, the lack of fit
was manifested in more vonths of zero demand thdn that
described by the Poisson distribution. It was also true
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that we were getting more variation than was permitted
under the Poisson assumption. For these reansons and
others that will be discussed later, we used a distribution
known as the negative binomial to Ift the observed data.
It is a two-parameter distrihbtion of a discrete variable.
The follow, ng example illustrates this distribution.
The monthly deimand data fronm a squaeron of aircraft for a
door asse-mbly shows the following series of deinands over
a 36--month per 4 od: O-l-0--O---O-O-4-4-0-O.-, 0-0-0-1-0-0-0-
0-0-0-3-0-, 0-0-4-0-1-0-04---0-2-0-0. The sum of the de-
mands is 12. The mc-an demand per month is one-third. The
assumption of a Poisson distribution, using 1/3 as the
parameter Nialue, does not give a good fit to the data. The
negative !Anomial distribution, using the calculacion of
moments of tile obseived distribhtion to estimate che param-
eters. gives a good fit to the data. The ratio of variance
to neau• is *bout 2.•. If we uste an arbitrary ratio of
variance to mean of ',. the fit is also good. The inter-
pcetation of "good fit' means that the amount of discrepance
between the observed values and tt.e theoretical values
based .an the assumed distribution is not large enough to
.ndicate the presence of anything more than the capt.ces
of random sampling. In other words, the hypothesis is
upheld that this data could have been obtained from a
population in which monthly demand was described by a
negative binomial distr.bution.
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The negative bioK.mial distribution is completely
defined by two parameters. the arithmetic mean m and a
positive exponent k. The distribution is written
(q - p)-k where p - n/k and p + 1 - q. The general term
in the expansion of this binomial gives the probability
P that an observation x will have values 0, 1, 2, "'.
The general term may be written
P(x) I _+_' x - 0,1,2,..., P, k > 0." (k--l)-"xI. q k+X ""
The curve defined by the value of P(x) is unimodal, so
that in fitting the negative binomial to an observed
distribution, any apparent bimodal or multimodal tendency
is attributed to random sampling. The negative binomial
is an extension of the Poisson series in which the popula-
tion mean m is not constant but varies contin, wusly in a
distribution which is proportional to that of Chi-squarr
(The distribution referred to is called Pearson Type III
or Gauma distribution). Thus the negative binomial may be
used to represent a composite of several Poisson distribu-
tions in which the number of observations per unit time
in repeated counts cannot be assumed to have the same
expected valut mean) in each unit of measurement. Student
[12] wrote in 1919 as follows: "If the presence of one
individual in a division increases the chance of other
individuals falling into that division, a negative binomial
will fit best, but if it decreases the chance, a positive
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binomial." Bliss reported [13] on fitting the negative ,
binomnial to biological data. "The negative binomial is
thc easiest to compute and the most widely applicable of
the distributions for over-dispersion."
The negative binomial has been used in a variety of
applications to biological data. It was used by Bliss 13)
to fit a distribution function to counts of red mites on
apple leaves. It was used by Morgan et al [14] to describe
the distribution of bacterial clumps over a milk film.
Student [15] used it for a description of the counting of
yeast cells with a haemocyto.Teter. Stirritt [161 et al.
foud that it was adequate to describe the distribucion of
corn borers in a field experiment. Greenwood and Yule [17]
used a negative binomial to describe the dis~rib'rb on of
accidents experienced by machinists. (The theory was that
some machinists were more accident-prone than others.
There was aiso the possibility that shop conditions differed
from week to week in such a way as to cause accident risks
to vary during successive weeks of the period covered by
the data. In either case, the resulting distribution
mnight be expected to be of the fori of a negative binomial.)
In plant ecology, quadrant counts which deviated from the
Poisson distribution were attributed to the occurrence of
plants in "clumps." Blackman [18] found that the distri-
bl:tLon of plants per quadrant agreed very well with a
negative binomi.al distribution, with estimates of param-
ezers derived from the sample data. Jones, Mollison, and
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Quenouille [19) used the negative binodal in fitting
counts of soil bacteria. Sichel (20] made use of a
negative binomial in studying psychological data on the
occurrence of minor accidents in an industrial plant.
Additional references could be cited, but our purpose
is merely to document the fact that the negative binomial
distribution has been used for years in widely divergent
fields of applicationz.
Here we are interested in the application of this
particular distribution to the demand for aircraft spare
parts. We shall first discuss the rationale which leads
us to believe that it may offer a reasonable description
of spare-parts demand.
Suppose that two-years' data on demand for spare
parts is available at a base. For illustrative purposes,
we will say that it is a bass with four squadrons, each
consisting of 18 aircraft. that is, the data covers the
demands of 72 planes for 24 months. If all the aircraft
were Identical, of the san ape, flew identical missions,
underwent identical servicito, -ere subject to the sawe
maintenance practices, etc., the demand for actuator parts
might be expected to be about th.e same as in the illustra-
tive example on page 4. In this case, the monthly demands
would follow the Poibson distribution. If the population
were homogeneous, random samples of demand data vould be
expected to exhibit only the sampling fluctuations in-.
herent in any ve1l-behaved variable. But the homogeneity
described above is not present in the Air Force environment
in which the demands are generated. For example, even
though the planes fly equal numbers of hours, make the same
number of landings, etc., some planes are nevertheless
subject to greater stresses than others because of factors
of speed, altitude, rate of climb, etc. The probability
of failure of a particular part is no longer constant, but
has increased due to the stre3s factor. The recorded
demands at the base for this part over the 24-month period
might be as follows: 1-0-7-0-0-3-1-2-0-0-6-I-, 0-1-0-4-1-1-
2-0-9-2-0-3. The sun, of the dcmuands at the base is 44.
There arc four squadrons, so that the demand rate per
squadron month is the same as in the Poisson-fitted example
on p. 4.
For these demand data, the Poisson is not a good fit.
If we use the X2 statistic to test the egreement of the
observed distribution with the theoretical Paisson distri-
bution, the computed X2 value for I d.f. is 8.3. The
probability of obtaining a X 2 value of this nasnit.ade or
greater in drawing random samples from a homogeneous
population is less than .01. Such large values of chi-
square may signify no more than the presence of an tun.-
usually divergent sample, but to the investigator who is
none too sure of his hypothesis, the presence of repeated
large chi-square values indicates that the hypothesis
should be rejected.
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However, if we usc the negative binomial probability
distribution with the same mean demand per month ind assume
that the ratio of variance to mean is 3, we will obtain a
good fit for the sample data. (G.2 is .17 for 1 d.f.,
P = .70.) The discrepance between the observed frequency
distribution of demands per month and tho theoretical
frequencies based on the negative binomial is v'ery small
indeed. The ratio of sample variance to sample mean in
this case was about 3.2. A word of cauticn may be appro-
priate here concerning inferences made about the population
on the basis of the chi-square test of goodness of fit.
Th. viewpoint adopteO is that the hypothesis is fixed-
namely, that a negative binomial distribution describes
the population of monthly demand..s fur the part. The
probability evaluated above is that of drawing from the
population a sample more extreme than the one in hand. It
is not a method for evaluating the probability that the
hypothesis is correct. Of course, it is true that after
the evidence from samples is accumulate. the analyst-
researcher must make a decision about the hypothesis, and
his decision has some presumably high probability of being
correct; but we have not presented a method of evaluating
such a probability.
There are many factors in the Air Force environment
that ccntribute to heterogeneity of demands. Among these
factors we might list age of aircraft, applicability of
parts, flying-ipogram elements, nature of mission,
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servicing schedules, maintenance practices, design change
and modifications, changes in base personnel, etc. A
crew mechanic inspecting plane number 1 may not have the
same careful standards as the mechanic who inspects number
2. Some inspectors are prone to replace parts-some are not.
Demands for certain spare parts have also been known to
exhibit a tendency to cccur in clusters--that is, the demand
for a unit of part a stimulates demand for a unit of part b,
which in turn creates a demand for 4 units of part c, etc.
All of these factors change with time, and are reflected in
the sample data.
4. TABLES OF NEGATIVE BINOKL DISTRIBUTION
We have found tables of the negative binomial distri-
bution useful in our work in logIstics. Since they may
not be readily accessible, the complete probability distri-
butions for a limLted number of arbitrary parameter values
are presented in the following paps. We have used 13
different values of the man (a a kp) and 10 different
ratios of varianc*-Co--man (q). This XLves us 130 sets
of parameter values for which the complete probability
distribution in tabulated.
The distribution function of the negative binomial is
P(x) - I
(k-)O. X0 q
where x = 001,203--, pok > 0, and q a 1 + p.
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Part I gives the individual term of the distributions
and Part 2 gives the cumulative probability of x demands
co): less. The values of the mean (kp) are .25(.25)1.0 and
1.0(-,0)10.0. The values of the ratio of variance to mean
(q) zre 1.5(.5)5.0 and 5.0(1.0)7.0. The choice of these
values makes p vary from 1/2 to 6 and k from 1 to 20.
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REFERENCES
1. Petersen, J. W., Savings from Procurement Deferralwith Interim Contractor Support: The Case of HighValue Airframe spares, The RAND Corporati'in,RM-ZU85, January 1,U 1958.
2. Karr, H., B. Brown, and M. A. Geisler, A PreferredMethod for Designing a Flyaway Kit, Te RANDCorporation, KM-14VO, May 19, 1955 (Confidential).
3. Fort, D. M., Experimental Design and Evaluation of anF--86H Flyaway Kit, The RAND Corporation, R1--2 2,7December 16, 1977.
4. Goldman, T. A., A Priori Demand Prediction-A CaseStudy of B-52 Airframe Parts, The RAND Corporation,RM--2C)5, Januar'y IU, 1958.
5. Geisler, M., and B. Brown, Analysis of Demand Patternsfor "-47 Parts at Base Level, The RAND Corporation,R.M--1297, July 1957.
6. Ferguson, A. R., Efficient Stockage of Category IIand III Parts, The RAND Corporation, 5-23, October
=2, 7T97.7. Ferguson, A. R., and L. Fisher, Stockage Policies for
Medium and Low Cost Parts, The RAND orporation,RM-19b2, April 15, 1955.
8. Petersen, J. and M. Geialer, Cost of Various BaseStockage Policies, RI-1392, December 5, 1954.
9. Molina, E. C. Poisson's Exponential "Binomial Limit,D. Van Nostrand, 1942.
10. Kitigawa, T., Tables of Poisson Distribution, Baifukan,Tokyo, Japan, 775Z.
11. Pearson, E. S. and H. 0. Hartley, Biometrika Tablesfor Statisticians, Vol. 1, Caarrdge universityPress, It Y4.
12. Student,'Anpila.ion of Deviations from Poisson'sLaw in Pr••ices," lometrika IZ, IVLV, pp. 211-215.
13. Bliss, C. I. and R. A. Fisher, "Fitting the NegativeBinomial Distribution to Biologicai Data and Noteon the Efficient Fitting of the Negativc Binomial,"Biometrics 9, 1953, pp. 176-200.
14. Margan, M. E., P. MacLeod, E. 0. Anderson, and C. I.BfIss, "A Sequential Procedure for Grad-ing Milk byMicroscopic Counts," Storrs, Conn. AMr. Exp. Sta.Bulletin, 1951, p. 276.
15. Student, "On the Error of Counting with a Haemocytometer,"Biometrika, 5, 1907, pp. 351-360.
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16. Stirrict, G. M., G. Beall, and M. Timonin, "A FieldExperiment on the Control of the European CornBoret," Sctentific Agriculture 17, 1937, pp. 587-591.
17. Greenwood, M.. and G. U. Yule, "An Inquiry into theNatutre of Frequency Distributions Representative ofMultlple Happenings," Journal of the Royal Statisti-calz So Riet, 83, 1920, pp. 253-279.
18. Blackr~an, G. El., "A Study by Statistical Methods ofthe Distribution of Species in Grassland Associations,"Annails of Bctany 49, 1935, p. 749.
'9. Jones, P.C.T., J. E. MoLlis,n and M. H. Quenouille,"A Technique for the Quantitative Estimation ofSoil Microorganisms," Journal of Genetics and Micro-b__logy 2, I__ 48, pp. 54--19.
2G. Sichol, Herbert S., "The Estimation of the Parametersof a Negative Binomial Distribution with SpecialReference to Psychological Data," Psychometrika 16,1951, pp. 107-126.