VOL. 2, No. 4 TECHNOMETRICS NOVEMBER, 1960
On the Evaluation
of the Negative Binomial
Distribution
with Examples
G. P.
PATILIndian Statistical Institute and The University oj Michigan
The Negative Binomial Distribution is a frequently encountered standard dis- Crete distribution function. By the use of the available Binomial or Incomplete Beta Function tables it can be evaluated easily through the use of certain identities.
1. INTRODUCTION
Different approaches are possible with regard to the basic structure of the standard discrete distributions like the binomial, Poisson, negative binomial, or logarithmic series. Under plausible circumstances, these distributions may be regarded as descriptive models of populations. For instance, the number of accidents met with, over a period of time by a particular individual, can ordinarily be assumed to have a Poisson distribution. However, different persons may have different accident proneness as measured by the average number of accidents to the individual. If this average has, say, a Pearson’s type III distribution, the distribution of the number of accidents pooled over the individuals can be shown to follow the negative binomial law.
Again, the standard discrete distributions may arise as a result of the sampling scheme adopted. In sampling with replacement n items from a lot of manu- factured items, the number of defectives follows the binomial law. On the other hand, if one uses what is known as the inverse binomial sampling procedure (that is, one goes on sampling with replacement until he get a fixed number such as k of defectives), the number of items sampled follows the negative binomial distribution.
Let
2. SUMMARY
YhPP,k) =(k+;-l)p’(l -p)”
(1)
whereO<p<l,O<Ic< m,x=O,1,2,....In order to evaluate the negative binomial distribution function:
i-0
we can use (positive) binomial distribution function tables, when lc is a positive integer, or tables of the incomplete beta function for any k. Thus no special tables are needed, since:
Y(r, P, Ic) = 1 - B(k - 1, P, T + U,
501
where: and
W-, P, W = I,@, r + 11,
O<k<m where : 1 v Ip(m, n) = ___am, 4 s 0 Um-l(l - u)“-l du
(4)
(5)
(the incomplete beta function)
3. PROOFS
The proof of (3) is as follows:
Y(r,
P, k> = 2 (” + ; - ‘)p’(l - p)”
= Probability that at most k + r independent trials are required to get k successes when p is the probability of success at each trial.
= Probability that at least k successes occur in k + r independent trials when p is the probability of success at each trial.
= g (” ; qpyl _ p)k+r--2
= 1 -B(k- l,p,r+k).
The proof of (3) given above is based upon probability argument. The identity is essentially algebraic and hence it appears to be interesting to establish it algebraically. The following alternative algebraic proof of (3) is due to Professor C. C. Craig.
The identity to be established is k+x-1
z )PV
Divide by pk and then set 1 - p = u. The identity becomes
k(
r-0
k+,-l)uz = g(z’;$ -U)=UT-=
In the right hand side of (7) the coefficient of u*(O I s i r) is
EVALUATION OF NEGATIVE BINOMIAL DISTRIBUTION
This is the coefficient of U* in
503
(-I)“(1 - U)r+k(l - U)-(--a+l) = (1 - U)k+a-‘(-l)‘
But the coefficient of u8 in the left hand side of (7) is
( k + ’ - ’ s > - Hence the identity. Also we have 1 p L(k, r + 1) = B(k, r + 1> I o u k--l(l - ZL)’ du
f Cl,@,
r + 01 = B(k, f + 1) pk-'(' - d'
=
(1 - PI’ = ji dr, p, k + 1) Next, $ [Y(r, p, k)] = 2 (” + ; - ‘)[kpk-‘(1 - p)O>” - q+(l - p)“+]z-0
=
2 (” + ; - ‘)p”‘[k(l - p)O>” -t ~(1 - p - l)(l - p)“”z-0
=Z 2 (” + ; - ‘)p”-‘[(k + x)(1 - p)= - ~(1 - $-‘I2-O
= Hence = $ [Y(r,p, k + l)] - $ [Y(r - l,p, k + l)] = $/h P, k + 1) $ [Yh-, p, k)] = 2 [I,(k, r + l)]integrating (10) with respect to p, we have
Y(r, p, k) = I,(k, r + 1) -I- C where C is a comkant of integration.
Nowatp = 0, Y(r,O,k) = 0 = I,(k,r+ 1) * C = 0 and hence the statement (5). . .
(9)
(10)
4. APPLICATIONS
1. The results obtained above can be used to compute with ease the expected frequencies when one is trying to fit the negative binomial distribution. After the parameters p and k are suitably estimated, all that one need do is to refer to the Incomplete Beta Function tables (or to the Binomial Distribution tables) for evaluating the expected cumulative frequencies and hence the expected frequencies.
2. The results can be utilized in solving acceptance sampling problem of the following type. Suppose that an acceptance sampling plan calls for drawing (with replacement) units from a lot until one get 5 defectives. If 50 drawings or less are required, one rejects the lot. If more than 50 drawings are required, one accepts the lot. What is the probability of accepting a lot that is 10 per cent defective?
Writing k = 5, p = .lO and N = 50, we have the probability of acceptance given by
pa = j!zl (; 1 ; PYl - PY >
= 1 - x (” + ; - I)p”(l - p)” s
= l- Y(N - k, P, kl
By (3), P, = B(k - 1, p, N) = R(4, .lO, 50). From Binomial tables, one has P. = 0.431199. More generally, we have the following.
3. A single-sample binomial sampling plan is characterized by the sample size n that should be taken and the acceptance number c of defective units that cannot be exceeded without the lot’s being rejected. The specifications of the plan are taken to be the producer’s risk CY at the acceptable quality level (A&L) and the consumer’s risk /3 at the lot-tolerance-fraction-defective (LTPD). These specifications dictate the particular choice of the parameters n and c of this plan determined by the two equations:
B(c, A&L, n) = 1 - cy w B(c, LTPD, n) = p (13) A single-sample inverse binomial sampling plan is characterized by N and k where N is the smallest number of drawings required for accepting the lot when one keeps drawing until one gets k defectives. The specifications of the plan are taken to be the same as in binomial sampling plan. These specifications dictate the particular choice of the parameters N and k of the inverse binomial sampling plan determined by the two equations:
W - k, A&L, k) = a (14) Y(N - k, LTPD, k) = 1 - ,9 (15) which in turn reduce to
B(k - 1, A&L, iV) = 1 - (Y 06) B(k - 1, LTPD, N) = /3 (17)
EVALUATION OF NEGATIVE BINOMIAL DISTRIBUTION 505
It is interesting to see the two similar pairs of equations [(12), (13)] and [(16), (17)] and hence the connection between the two types of sampling plans as reflected in their parameters related by n = N and c = k - 1 which seems to make a lot of meaning even intuitively. This suggests that tables and charts available to obtain binomial single-sample plans can be used t,o obtain inverse binomial single-sample plans for given specifications. One has N = n and k=c+l.
REFERENCES
1. BLISS, C. I., “Fitting the Negative Binomial Distribution to Biological Data, with a Note on the Efficient Fitting of the Negative Binomial Distribution”, Biometrics, 9, 176-200 (1953).
2. CRAIO, c. c., “Note on the Use of Fixed Numbers of Defectives and Variable Sample Sizes in Sampling by Attributes”, Industrial Quality Control, 9, No. 6, 4345, (1953). 3. FISHER, R. A., “The Negative Binomial Distribution”, Annals of Eugenics, II, 182-187,
(1941).
4. GREENWOOD AND YUIX, ‘An Inquiry Into the Nature of Frequency Distributions Repre- sentative of Multiple Happenings with Particular Reference to the Occurrence of Multiple Attacks of Disease or of Repeated Accidents,” J.R.S.S., 83, 235-279, (1920).
5. HALDANE, J. B. S., “The Cumulants and Moments d the Binomial Distribution and The Cumulants of Chi-Square for a(n.r2)-fold Table,” Biometrika, $1, 392-395, (1939). 6. PATIL, G. P., Thesis for the Associateship of the Indian Statistical Institute, (1957). 7. PATIL, G. P., Ph.D. Thesis Submitted to The University of Michigan, (1959).
8. SICHEL, H. S., “The Estimation of the Parameters of a Negative Binomial Distribution with Special Reference to Psychological Data”, Psychcmetrika, 16, 107-127.
9. WISE, M. E., “The Use of the Negative Binomial Distribution in an Industrial Sampling Problem”, Supplement to J.R.S.S., 8, 202-211, (1946).
10. Applied Mathematics Series 6. “Tables of the Binomial Probability Distribution”, Na- tional Bureau of Standards, U. S. Govt. Printing Office, Washington, D. C. 1949. 11. PEARSON, K., Tables of the Incomplete Beta Function, Cambridge University Press, London.
