42 IEEE TRANSACTIONS ON RELIABILITY, VOL. R- 19, NO. 2, MAY 1970

Confidence Intervals for the Ratio ofTwo Poisson Means and Poisson

Predictor IntervalsWAYNE NELSON

Abstract-Easy-to-use charts are presented for confidence intervals for value X is an interval that encloses the future value Y withthe ratio of two Poisson mean occurrence rates and for predictor intervals probability y. A precise definition is given later. The ex-for the future value of a Poisson random variable, given a previously ob- pectations of Y and X are i.t and ).s, where i is an unknownserved value. The use of the charts is illustrated and their theoretical basisis gien common mean occurrence rate and t and s are known

lengths of observation. This formulation of the modelReader Aids: includes the case where the mean occurrence rates have

Purpose: Present useful charts Purpose: Present useful chartsaknow rto23] and where X or Y is the sum of indepen-Special math needed for explanations: Elementary statistics, probability wn ratio [2theory dent Poisson random variables. Predictors and predictor

Special math needed for results: Elementary statistics, probability theory intervals for the future value Y are presented.Results useful to: Reliability engineers The next two sections comprise the body of this paper.

The first illustrates how to obtain an estimate and con-fidence interval for the ratio oftwo Poisson mean occurrencerates. The second illustrates how to obtain a prediction

T HE POISSON distribution has been used in a great and predictor interval for the future value of a Poissonvariety of applications [12] as a model for the number random variable. The appendixes contain theory for esti-

of occurrences of some event. Two such applications mation and confidence intervals and for prediction and[1], [11] encountered in the author's consultation stimu- predictor intervals and a description of the preparation oflated him to develop the charts given here. They have two the charts.uses: 1) to obtain confidence intervals for the ratio of twoPoisson mean occurrence rates, and 2) to obtain predictorintervals for the future value of a Poisson random variable, ESTIMATION AND CONFIDENCE INTERVALSgiven a previously observed value. These confidence and The method for obtaining confidence intervals for thepredictor intervals are based on the following statistical . of two Poisson mean occurrence rates is illustrated

models. ~~~~~~~~~~~raw1 ftoPlsnma curnerts1 lUtaemodels. by the following example [11], which motivated the author'sThe model underlying the confidence intervals is the work on this problem.

following. Let X be a Poisson random variable with mean The failure rates of two types of powerline wire in use inas, where . is an unknown mean occurrence rate and S is the same geographical region are to be compared. it isthe known length of observation. Similarly, let Y be a assumed that the number of line failures for a given exposurePoisson random variable, statistically independent of X, is a Poisson random variable. The standard bare wire hadwith mean ,ut, where , is an unknown mean occurrence rate X =69 failures ins = 1079.6 thousand foot-years of ex-and t is the known length of observation. This formulationof the model includes the case where X or Y is the sum of posure, anda hovearete wire hadsfailures in t = 467.9 thousand foot-years of exposure.independent Poisson random variables with a commonmean occurrence rate. Estimators and confidence intervals The observed failure rates are X/s = 69/1079.6

srelative 0.0639 failures per thousand feet per year for the bare wireare prsne o h ai p /hcsteand j = Y/t = 12/467.9 = 0.0256 failures per thousand

mean occurrence rate. feet per year for the tree wire. The observed failure rate ofThe model underlying the predictor intervals is the follow- th tre wir reatv totebr iei h ai 1the tree wire relative to the bare wire iS the ratio P-= /ing. Let Y be a future value of a Poisson random variable, 0

and et Xbeaprevousl obsrvedvalu of Poison40 percent of that for bare wire. This key figure iS a naturalrandom variable idependent of Y. Losely speaking,' estimate of the relative performance of the two types of

lOGy-percent predictor interval based on the previouswieadsesytueinmkgancomccmpro.

As an aid to engineering judgments, confidence limitson the true relative failure rate are desired to provide an

Manuscript received September 22, 1969; revised December 5, 1969. incaonfthuceanyinhesime.CrsfrThe author is with the General Electric Research and Development ncao fthuceany hestme.C rsfr

Center, Schenectady, N.Y. such limits are given on log-log paper to permit necessary

NELSON: CONFIDENCE INTERVALS FOR RATIO OF POISSON MEANS 43

graphical multiplication and division with the same PREDICTION AND PREDICTOR INTERVALSoperations that are used on a slide rule. A one-cycle rule The methods for obtaining predictor intervals for ais provided with each chart and can be cut out and used to I

faciitat obann cofdnelmt. future Poisson observation are illustrated by the followingA two-saded 95-percent confidenceits.t i example, which is similar to a problem that motivatedA to-sded95-erentconidece ntrva fo th raiothe author's work. That problem iS described in [1], andoffailure rates is obtained from the chart for 95-percent two- the method ofs .Tuti is indcaed terie

sided intervals as follows. Enter the chart at the value of theratio Y =16o.T The powerline data indicate that the tree wire has a

rataio Y/Xlow12/69mit0.174donecthehorizontalmaxis.YTo lower failure rate than the bare wire. In view of this, in-

value to the lower curvegdiely wadrthe Y/X formation is desired for planning purposes about thevaluecestorythe loerpcurveolabeled wi heX values. 6n It number Y of line failures that would be observed in a yearis necessary to interpolate for some X values. Then gohorizontally to the vertical scale to read the value if the entire length, t = 515.8 thousand feet, of the powerline

were all tree wire. This calls for a predictor and predictorinterval based on the number X = 12 of failures previouslyobtain the lower limit. That is, the lower limit isobserved in s = 467.9 thousand foot-years of exposure.

p = (s/t)pl = (1079.6/467.9)0.087 = 0.20 or 20 percent. Note that the number of failures 12 was denoted by Yin the previous example.

The upper limit for the ratio is obtained from the upper The natural predictor for Y is Y* = it = tX/s =curve labeled with the X value, which is 69, by the same 515.8(12/467.9) = 13.2 failures. This predictor is the ob-method and is 75 percent. Thus the two-sided interval served failure rate A = X/s times the length t of obser-(20 and 75 percent) encloses the true relative failure rate vation for Y. It is best in a sense that is explained later.p = y/14Q with 95-percent confidence. Similarly, a one-sided A two-sided 95-percent predictor interval is one that97.5-percent upper confidence limit is 75 percent. encloses the future Y with 95-percent probability. SuchAn interval from the chart has a confidence level equal an interval is obtained from X as follows. Enter the approp-

to or greater than the stated value. This conservative riate chart at the value of the ratio t/s = 515.8/467.9 = 1.10property of the confidence intervals is a result of the discrete on the vertical axis. For the lower limit, go horizontally tonature of the Poisson distribution. the upper curve labeled with the X value, which is 12 here.The two-sided 95-percent confidence interval (20-75 It is necessary to interpolate for some X values. The value

percent) for the ratio does not enclose the value 100 per- 0.337 on the horizontal axis directly below this point oncent. For a two-sided statistical test of the null hypothesis of the curve must be multiplied by the X value to obtain theequal occurrence rates (p = 1), this corresponds to rejection lower limit Y = 0.337 x 12 = 4.04. The upper predictorof the null hypothesis at a 5-percent significance level. limit for Y is obtained from the lower curve labeled withMore generally, such a confidence interval can be used to the X value by the same method and is Y = 26.0. Thus thetest the null hypothesis that the true ratio p equals a two-sided predictor interval (4.04, 26.0) encloses the futurespecified value po. value Y with 95-percent probability. If desired, the lowerA confidence interval for a ratio is more informative limit can be rounded up and the upper limit rounded down

than a hypothesis test, since a confidence interval indicates to the nearest integer. The predictor interval then includesthe precision of the information in the data and a hypothesis the integer endpoints. For example, with tree wire, thetest does not. For an economic comparison of the two types powerline would have in a year from 5 to 26 failures withof wire, it would not be enough to know that their failure 95-percent probability. Similarly, a one-sided 97.5-percentrates differ significantly. One would need to know with upper predictor limit is 26 failures.high confidence that the failure rate for one type of wire is A predictor interval from the chart encloses the futureless than a certain fraction of the failure rate for the other. observation with a probability equal to or greater thanThis information is provided by a confidence interval. the stated value. This conservative property of these

If X and Y are large, the following procedure, which predictor intervals is a result of the discrete nature of thedoes not require the charts, provides approximate con- Poisson distribution.fidence limits for p. The powerline data illustrate the pro- If X and Y* are large, the following procedure, whichcedure. Calculate a' /X + 1/Y= /6 1/12 = does not require the charts, provides approximate predictor0.313. Then approximate lower and upper limits for a two- limits for Y. The powerline data illustrate the procedure.sided 100y-percent confidence interval for p are Calculate a* = /1/X + 1/Y* = -1/12 + 1/13.2 = 0.399.

Then approximate lower and upper limits for a lOO;y-p = p5/exp (z6r) and p3 = p5 exp (ziX) percent two-sided predictor interval for Y arewhere zis the 100(1 + ^y)/2 percentile ofthe standard normal Y-Y*/exp (zu*) Y- Y* exp (vo*)distribution. For the powerline data, approximate two-P P)sided 95-percent limits are p = 0.401/exp (1.96 .0.313) = where z is the 100(1 + y)/2 percentile of the standard0.22 and p-=0.401 exp (1.96 .0.313) =0.74. These limits normal distribution. For the powerline data, approximateare reasonably close to the previous ones. 95-percent predictor limits for Y are Y = 13.2/ exp (1.96.

44 IEEE TRANSACTIONS ON RELIABILITY, MAY 1970

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NELSON: CONFIDENCE INTERVALS FOR RATIO OF POISSON MEANS s

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46 IEEE TRANSACTIONS ON RELIABILITY, MAY 1970

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NELSON: CONFIDENCE INTERVALS FOR RATIO OF POISSON MEANS 47

0.399) = 6.0 and Y = 13.2 exp (1.96. 0.399) = 28.9. These CONCLUSIONSlimits are reasonably close to the previous ones.

It i ineresingto cmpae tis peditor nteval The easy-to-use charts given here provide confidenceI intervals for the ratio of two Poisson means and therebybased on the assumption that true mean occurrence rate A . .m o

is not known, with the predictor interval based on the almore inf rti ans of tesinthe hptesis ofassumptio tha iti.nw n qa oteosre equality. The charts also provide predictor intervals for aassumption that it iS known and equal to the observed

value of 0.0256 failures per thousand feet per year. This future Poisson observation.

knowledge corresponds to having observed the powerline APPENDIX Ifor an essentially infinite length of time, that is, s = oo. Then*= 1/,t = `1/13.2 = 0.275, and the lower and upper The theoretical basis for the charts and their preparation95-percent prediction limits are are briefly presented in the Appendixes. Greater detail is

Y = {t/ exp (zc*) = 13.2/ exp (1.96. 0.275) = 7.7 given in [19].

Y = it exp (zu*) 13.2 exp (1.96 0.275) = 22.7. Theoryfor Estimation and Confidence IntervalsThese limits are not much tighter than the previous ones. Presented here is the theory for estimators and confidenceThis is typical of prediction problems where there is a intervals for the ratio p = p/IA of two Poisson mean occur-reasonable amount of past data and indicates that the rence rates.variability in Y is comparable to or greater than the varia- The natural estimator for p = p/i is p - (Y/t)/(X/s),bility in At. The exact probability that this interval encloses that is, the ratio- of the usual estimators i = Y/t andY is P(7.7 < Y < 22.7) = 0.943 for a Poisson distribution 4 = X/s for p and a. p is the maximum likelihood estimatorwith mean equal to ).t = 0.0256 515.8 = 13.2. for p, and its asymptotic variance is p2[1/is + 1/it] for

s and t large. This estimator for p is biased, and ChapmanSome Remllarks on Prediction [6] has shown that no unbiased estimator exists. He

Traditional statistical analysis, in contrast to prediction proposes (Y/t)s/(X + 1), which is almost unbiased.analysis, is concerned with using the information in samples Confidence limits are based on the conditional distribu-to make inferences about the populations or distributions tion of Y, given X + Y = n, which is binomial [20] withfrom which the samples were obtained. Estimates, con- sample size n and parameter p = pt/(pt + is). Iffidence intervals, and tests of hypotheses are usually used p = p(cx; Y, n) and = P(o'; Y, n) are lower and upperto make such inferences. For purely scientific problems, confidence limits with confidence y = 1 a - c' for aanalyses that provide such statistical inferences are appro- binomial parameter p, thenpriate since the characteristics of the underlying populations Pr {p < pt/c(t + is) . i5 = (1)or distributions are usually a scientist's primary interest.On the other hand, for many statistical problems in business for any values of ). and pl. This confidence statement, whichand engineering, such as the previous example, one seeks is conditional on the total number n X + Y of occur-information from a sample that will provide guidance for rences, can be rewritten assome action. Then one is concerned with the outcome of a Pr {(s/t)p/(1 - p) < p/i . (s/t)j-/(- (2)future sample which will determine the consequences / -pof the action. In such situations, the underlying distribution which provides 1007-percent confidence limits for the ratiois not of interest in itself but is only a means of making p/). Any type [3], [7], [10], [22] of confidence limits for apredictions about a future sample. A general discussion of binomial proportion can be used in (2).prediction problems is given in [24] from a decision theory The Clopper-Pearson [7] limits for a binomial proportionpoint of view and in [18] from a classical statistics point were used in (2) in preparing the charts because they areof view. simple, nonrandomized, and well known. These limits are

It is appropriate to make two comments on the use of p = x(i; Y, X + 1) and p-= x(1- '; Y + 1, X), wherepredictor intervals. It is assumed that the mean occurrence x(x; Y, X + 1) is the 100cx percentile of the beta distributionrate A is the same for X and Y. Such an assumption of [14] with parameters Y and X + 1. Because the binomialstatistical stability underlies almost all statistical analyses distribution is discrete, these binomial limits and those forand must be critically examined in any application. In the the ratio are conservative in the sense that the probabilitypowerline example, year to year variation in weather with that they enclose the true value is at least 7. The confidencea corresponding variation in the mean failure rate of a statement in (2) can be written aspowerline would make the powerline prediction somewhat Pr {(s/t)p1(c&.; X, Y/X) . pu/i < (s/t)fi1(a~: X, Y/X)}in error [1], [11]._The interpretation of a predictor interval should be .7 1- cc-' (3)

made clear. A predictor interval applies to a pair of obser- where Pi(a; X, Y/X) -(Y/(X A- 1))/F(x; 2X -+ 2, 2Y) andvations where the first is used to obtain an interval for the j51(x'; X, Y/X) F(a', 2Y -+ 2, 2X)(Y +- 1)/X, and F(xL;second. The probability level 7 is the expected proportion 2X A- 2, 2 Y) is the upper 100&x-percentage point of the Fof such pairs for which the interval encloses the second distribution [17] with 2X +- 2 degrees of freedom in theobservation. A given predictor interval does not enclose numerator and 2Yinthe denominator. The charts give thean expected proportion 7 ofa number of future observations, functions p 1 and p for oc =oc

48 IEEE TRANSACTIONS ON RELIABILITY, MAY 1970

This development of the confidence limits relies on of the spread of the prediction error is the variancewell-known results and is more general and simpler than var{{Y*- Y} = varl{tX/s} + var {Y} = )t(s + t)/s. Thisothers in the literature [2], [4]-[6], [8], [9], which are sum- predictor is a uniformly minimum variance unbiasedmarized in [12]. Essentially the same development is predictor in the sense that its variance, var,{Y*- Y}, isgiven in [2], [6]. Cox [8] provides a simple heuristic deriva- less than or equal to the variance of any other unbiasedtion from a Poisson process point of view and obtains predictor for any value of i, and it is the unique best one inapproximate limits, based on F percentiles, similar to those this sense [18].given previously. Harris [13] gives a method for obtaining An interval with lower and upper endpoints Y _ y(X)confidence limits for the ratio of two products of any and Y =y(X), which are functions of the previouslynumber of Poisson means but does not indicate the simple observed value X, is called a two-sided 1007-percentsolution for the special case considered here. Lindley [16] predictor interval for the future value Y ifgives Bayesian confidence limits for gamma distribution PrA {y(X) < Y < y(X)} .priors for the Poisson means.The confidence interval given here for the ratio of the for any value of A,. That is, the interval based on the first

two mean occurrence rates is related to the standard observation encloses the second observation with proba-hypothesis test [2], [6], [8], [15], [20] for the equality of bility at least x. One-sided predictor intervals are definedtwo Poisson mean occurrence rates, as they are both similarly.based on the same statistic and sampling distribution. A derivation is given here for a one-sided PoissonSuch relationships between a confidence interval and a predictor interval. The derivation for a two-sided intervalhypothesis test exist for many problems in statistics, for is similar. The one-sided upper predictor limit for Yexample, the confidence interval and related hypothesis obtained from X by the graphical method is Y A-(X) =test based on an F statistic for the ratio of the variances of Xp '(y; X, t/s), where p7 (y;X, ) is the inverse of thesamples from two nornal distributions. function P1(7; X, ) which is graphed in the charts. Because

(s/t)p 1(y; X, Y/X) is a lower confidence limit for the ratio

AheapproximateC onfidenceInter valvalsn As and= 1 of the mean occurrence rates for X and Y it follows

The approximate confidence interval when iLs and ,ut thatare large is based on the fact that the natural logarithmof p(Xt/Ys) = p/l is approximately normally distributed , . PrA {1 . (sJt)po(>'; X, Y/X)}with mean zero and variance a2 = [1/is + 1/it]. This By the monotonicity of the function p1(7; X, ),the previousapproximation is similar to the Fisher z transform for an relation can be rewritten in terms of the inverse functionF statistic. With this approximation p7 1(7; X, -) as

Pr{-zo<.ln(p/l) <z}z ^Y 7.PrIY.Xp(y;X,t/s)}

where z is the 100(1 + y)/2 percentile of the standard normal which gives the upper predictor limit for Y. The interval isdistribution. Equivalently, conservative in the sense that the future value Y is enclosed

Pr {p/ exp (zo) < p < p exp (zo)} ^. with probability at least 7. This is a result of the discreteI 2 c 2 nature of the Poisson distribution and the conditionalThe variance a' can be estimated by a" = 1X + 1/Y. bnma itiuin

Thus p -/ exp (za) and _ p exp (za) are approximate. . p A search of the literature for work on the Poisson1007-percent confidence limits for a two-sided interval prediction problem revealed that Weiss [23], who developedfor p. an approximate predictor interval, was apparently the

APPENDIX II first to consider the problem. Subsequently, in [1], theauthor indicated the predictor interval based on the F

Theoryfor Predictors and Predictor Limits distribution and presented it in [19]. Shah [21] presents anPresented here is theory for predictors and predictor equivalent predictor interval based on the binomial

intervals for a future value Y of a Poisson random variable distribution.that are based on an independent previously observed The predictor interval given here for a future Poissonvalue X. For a general presentation of predictors and observation is related to the confidence interval for thepredictor intervals and methods for obtaining them, see ratio of two mean occurrence rates and the hypothesis[18]. test for equality of two mean occurrence rates, as they areA predictor Y* for the future value Y is a function all based on the same statistic and sampling distribution.Y*= y*(X) of the previously observed independent value That a predictor interval can be obtained for many two-

X. The prediction error Ye-Yisa random quantity and, sample problems by inverting a confidence interval ortherefore, has a sampling distribution which should be hypothesis test is not widely recognized. For example, acentered on zero and have a small spread. predictor interval for the variance of a second independentThe predictor Y* = tX/s for Y is unbiased in the sense sample can be obtained from the variance of a first sample,

that the expected value of the prediction error is zero, where both are from the same normal distribution, wherethat is,E {Y*-Y} = ER{X/s} - E{Y} = tXs/s - t = 0, the interval is based on the F distribution for the ratio offor all values of A. For the unbiased predictor, a measure the two variances.

NELSON: CONFIDENCE INTERVALS FOR RATIO OF POISSON MEANS 49

Approximate Predictor Intervals REFERENCESThe approximate predictor interval for large A's and At [1] P. F. Albrecht, W. Nelson, and R. J. Ringlee, Discussion of A. D.

is based on the fact that the natural logarithm of Patton, "Determination and analysis of data for reliability studies,"IEEE Trans. Power Apparatus and Systems, vol. PAS-87, pp. 84-100,Xt/Ys = Y*/Y is approximately normally distributed with January 1968.

mean zero and variance a2 = 1/i;s + 1/it, which is es- [2] A. Birnbaum, "Statistical methods for poisson processes and ex-timated by Ur*2 = 1/X + 1/Y*. With this approximation ponential populations," J. Am. Statist. Assoc., vol. 49, pp. 254-266,

1954.Pr {-za* . ln (Y*/ Y) < za*} [3] C. R. Blyth and D. W. Hutchinson, "Table of Neyman-shortest un-

biased confidence intervals for the binomial parameter," Biometrika,where z is the 100(1 + y)/2 percentile of the standard vol. 47, pp. 381-391, 1960.

normaldistribution. Equivalently, [4] L. N. Bol'shev, "Comparison ofparameters ofPoisson distributions,"normal distribution. Equivalently, Theory Probability Appl., vol. 7, pp. 113-114, 1962.

[5] I. Bross, "A confidence interval for a percentage increase," Bio-Pr { Y*/ exp (zc*) < Y . Y* exp (za*)} % T metrics, vol. 10, pp. 245-250, 1954.

[6] D. G. Chapman, "On tests and estimates for the ratio of PoissonThus Y _ Y*/ exp (zu*) and Y- Ye exp (zu*) are ap- means," Ann. Inst. Statist. Math. (Tokyo), vol. 4, pp. 45-49, 1952.proximate 100y-percent predictor limits for a two-sided [7] C. J. Clopper and E. S. Pearson, "The use of confidence or fiducialinterval for Y. A more accurate and computationally limits illustrated in the case of the binomial," Biometrika, vol. 26,laborious approximation is given in [18], [23]. []pp. 404-413, 1934.l8] D. R. Cox, "Some simple approximate tests for Poisson variates,"

Biometrika, vol. 40, pp. 354-360, 1953.[9] D. R. Cox and P. A. Lewis, Statistical Analysis of Series of Events.

APPENDIX III London: Methuen, 1966.[10] E. L. Crow, "Confidence intervals for a proportion," Biometrika,

Preparation of the Charts [11]vol. 43, pp. 423-435, 1956.

FI1] E. S. Gardner, Jr., and R. J. Ringlee, "Line stormproofing receivesThe functions P, and -1 graphed in the charts were critical evaluation," Transmission and Distribution, vol. 20, pp. 59-61,P ~~ ~ ~ ~~lcrc June 1968.computed with a revised version of the General Electric [12] F. A. Haight, Handbook of the Poisson Distribution. New York:

Time-Sharing library program BETA for the incomplete Wiley, 1967.beta function and its inverse. This revision was developed [13] B. Harris, "Hypothesis testing and confidence intervals for productsatthe author's request by Dr. G. Roe, of the General and quotients of Poisson parameters with applications to reliability,"at the author's request by Dr. G. Roe, of the General Mathematics Research Center, University of Wisconsin, Madison,

Electric Research and Development Center, and gives MRC Tech. Rept. 943, August 1968; J. Am. Statist. Assoc. (to beaccurate percentiles over a wide range of parameter published).values. This program is documented in [251, but the limita- [14] H. L. Harter, New Tables of the Incomplete Gamma Function Ratio

and of Percentage Points of the Chi-Square and Beta Distributions.tions described there do not apply to the revised version. Washington, D.C.: U.S. Government Printing Office, 1963.The results of these computations were spot checked [15] P. G. Hoel, "Testing the homogeneity of Poisson frequencies," Ann.

against te Ftabls of Merington an Thompso [17][IMath. Statist., vol. 16, pp. 362-368, 1945.against the F tables of Merrington and Thompson [17] [16] D. V. Lindley, Introduction to Probability and Statistics from aand found to be in agreement with the five figures given Bayesian Viewpoint, Pt. 2: Inference. Cambridge, Mass.: Cambridgein their tables. It was necessary to use the BETA program Univ. Press, 1965.

[171 M. Merrington and C. M. Thompson, "Tables of percentage pointsas existing beta and F tables, e.g., [14], [17], are not extensive of the inverted beta (F) distribution," Biometrika, vol. 33, pp. 73-88,enough for the charts. The values of the functions were 1943.calculated for integer values of X and Y and plotted on [18] W. B. Nelson, "Two-sample prediction," General Electric Research

and Development Center, Schenectady, N.Y., TIS Rept. 68-C-404,log-log paper, and smooth curves were drawn through the November 1968.plotted points to obtain the charts. The curves can be [19] , "Confidence intervals for the ratio of two Poisson means andread to an accuracy of two or three figures, which is suf- predictor intervals for a Poisson random variable," General Electric

ficint frprcticl aplicaions Theauthr wil prvideResearch and Development Center, Schenectady, N.Y., TIS Rept.ficient for practical applications. The author will provide 69-C-1 18, February 1969.

on request double page size copies of the charts with a [20] J. Przyborowski and H. Wilenski, "Homogeneity of results ingreater range for X/Y running from 0.01 to 100. testing samples from Poisson series," Biometrika, vol. 3, pp. 313-323,

1940.[21] B. V. Shah, "On predicting failures in a future time period from

known observations," IEEE Trans. Reliability (Short Notes), vol.ACKNOWLEDGMENT R-18, pp. 203-204, November 1969.

[22] W. L. Stevens, "Fiducial limits of the parameter of a discontinuousThe author wishes to thank Dr. R. L. Shuey ofthe General random variable," Biometrika, vol. 37, pp. 117-129, 1950.Electric Research and Development Center for support [23] L. Weiss, "A note on confidence sets for random variables," Ann.and encouragement, Dr. G. Roe for providing the program Math. Statist., vol. 26, pp. 142-144, 1955.fortheincomplete beta function used in the calculations [24] --, Statistical Decision Theory. New York: McGraw-Hill, 1961.

[25] "Numerical analysis routines," Desk Side Time-Sharing Service,for the charts, and also P. F. Albrecht, G. J. Hahn, and Dr. General Electric Information Service Dept., Bethesda, Md., Publ.R. Ringlee for their stimulating suggestions. 805222, May 1968.