The Poisson probability law has become increasingly important in recent years as more and more random phenomena to which the law applies have been studied. In physics the random emission of electrons from the filament of a vacuum tube, or from a photosensitive substance under the influence of light, and the spontaneous decomposition of radioactive atomic nuclei lead to phenomena obeying a Poisson probability law. This law arises frequently in the fields of operations research and management science, since demands for service , whether upon the cashiers or salesmen of a department store, the stock clerk of a factory, the runways of an airport, the cargo-handling facilities of a port, the maintenance man of a machine shop, and the trunk lines of a telephone exchange, and also the rate at which service is rendered , often lead to random phenomena either exactly or approximately obeying a Poisson probability law. Such random phenomena also arise in connection with the occurrence of accidents, errors, breakdowns, and other similar calamities.
The kinds of random phenomena that lead to a Poisson probability law can best be understood by considering the kinds of phenomena that lead to a binomial probability law. The usual situation to which the binomial probability law applies is one in which
We next indicate some conditions under which one may expect that the number of occurrences of a random event occurring in time or space (such as the presence of an organism at a certain point in 3-dimensional space, or the arrival of an airplane at a certain point in time) obeys a Poisson probability law. We make the basic assumption that there exists a positive quantity
(i) the probability that exactly one event will occur in the interval is approximately equal to
(ii) the probability that exactly zero events occur in the interval is approximately equal to
(iii) the probability that two or more events occur in the interval is equal to a quantity
The parameter
Example 3A. Suppose one is observing the times at which automobiles arrive at a toll collector’s booth on a toll bridge. Let us suppose that we are informed that the mean rate
In addition to the assumption concerning the existence of the parameter
We now show, under these assumptions, that the number of occurrences of the event in a period of time (or space) of length (or area or volume)
Consequently, we may describe briefly a sequence of events occurring in time (or space), and which satisfy the foregoing assumptions, by saying that the events obey a Poisson probability law at the rate of
Note that if
To prove (3.1), we divide the time period of length
Now (3.2) is only an approximation to the probability that
as
It should be noted that the foregoing derivation of (3.1) is not completely rigorous. To give a rigorous proof of (3.1), one must treat the random phenomenon under consideration as a stochastic process. A sketch of such proof, using differential equations, is given in section 5.
Example 3B. It is known that bacteria of a certain kind occur in water at the rate of two bacteria per cubic centimeter of water. Assuming that this phenomenon obeys a Poisson probability law, what is the probability that a sample of two cubic centimeters of water will contain (i) no bacteria, (ii) at least two bacteria?
Solution
Under the assumptions made, it follows that the number of bacteria in a two-cubic-centimeter sample of water obeys a Poisson probability law with parameter
Example 3C. Misprints. In a certain published book of 520 pages 390 typographical errors occur. What is the probability that four pages, selected randomly by the printer as examples of his work, will be free from errors?
Solution
The problem as stated is incapable of mathematical solution. However, let us recast the problem as follows. Assume that typographical errors occur in the work of a certain printer in accordance with the Poisson probability law at the rate of
Example 3D. Shot noise in electron tubes. The sensitivity attainable with electronic amplifiers and apparatus is inherently limited by the spontaneous current fluctuations present in such devices, usually called noise. One source of noise in vacuum tubes is shot noise, which is due to the random emission of electrons from the heated cathode. Suppose that the potential difference between the cathode and the anode is so great that all electrons emitted by the cathode have such high velocities that there is no accumulation of electrons between the cathode and the anode (and thus no space charge). If we consider an emission of an electron from the cathode as an event, then the assumptions preceding (3.1) may be shown as satisfied (see W. B. Davenport, Jr. and W. L. Root, An Introduction to the Theory of Random Signals and Noise , McGraw-Hill, New York, 1958, pp. 112–119). Consequently, the number of electrons emitted from the cathode in a time interval of length
The Poisson probability law was first published in 1837 by Poisson in his book Recherches sur la probabilité des jugements en matière criminelle et en matière civile . In 1898, in a work entitled Das Gesetz der kleinen Zahlen , Bortkewitz described various applications of the Poisson distribution. However until 1907 the Poisson distribution was regarded as more of a curiosity than a useful scientific tool, since the applications made of it were to such phenomena as the suicides of women and children and deaths from the kick of a horse in the Prussian army. Because of its derivation as a limit of the binomial law, the Poisson law was usually described as the probability law of the number of successes in a very large number of independent repeated trials, each with a very small probability of success.
In 1907 the celebrated statistician W. S. Gosset (writing, as was his wont, under the pseudonym “Student”) deduced the Poisson law as the probability law of the number of minute corpuscles to be found in sample drops of a liquid, under the assumption that the corpuscles are distributed at random throughout the liquid; see “Student”, “On the error of counting with a Haemocytometer”, Biometrika , Vol. 5, p. 351. In 1910 the Poisson law was shown to fit the number of “
Although one is able to state assumptions under which a random phenomenon will obey a Poisson probability law with some parameter
be the total number of events observed in the
If we believe that the random phenomenon under observation obeys a Poisson probability law with parameter
Example 3E. Vacancies in the United States Supreme Court. W. A. Wallis, writing on “The Poisson Distribution and the Supreme Court”, Journal of the American Statistical Association , Vol. 31 (1936), pp. 376–380, reports that vacancies in the United States Supreme Court, either by death or resignation of members, occurred as follows during the 96 years, 1837 to 1932:
| 0 | 59 |
| 1 | 27 |
| 2 | 9 |
| over 3 | 1 |
Since
The foregoing data also provide a method of testing the hypothesis that vacancies in the Supreme Court obey a Poisson probability at the rate of 0.5 vacancies per year. If this is the case, then the probability that in a year there will be
The expected number of years in
| Number of Years out of 96 in which | |||
|---|---|---|---|
| 3-4 Number of Vacancies | Probability | Expected Number | Observed Number |
| 0 | 0.6065 | 58.224 | 59 |
| 1 | 0.3033 | 29.117 | 27 |
| 2 | 0.0758 | 7.277 | 9 |
| 3 | 0.0126 | 1.210 | 1 |
| over 3 | 0.0018 | 0.173 | 0 |
The observed and expected numbers may then be compared by various statistical criteria (such as the
The Poisson, and related, probability laws arise in a variety of ways in the mathematical theory of queues (waiting lines) and the mathematical theory of inventory and production control. We give a very simple example of an inventory problem. It should be noted that to make the following example more realistic one must take into account the costs of the various actions available.
Example 3F. An inventory problem. Suppose a retailer discovers that the number of items of a certain kind demanded by customers in a given time period obeys a Poisson probability law with known parameter
Solution
The problem is to find the number
The solution
Theoretical Exercises
3.1. A problem of aerial search. State conditions for the validity of the following assertion: if
3.2. The number of matches approximately obeys a Poisson probability law. Consider the number of matches obtained by distributing
Exercises
State carefully the probabilistic assumptions under which you solve the following problems. Keep in mind the empirically observed fact that the occurrence of accidents, errors, breakdowns, and so on, in many instances appear to obey Poisson probability laws.
3.1. The incidence of polio during the years 1949–1954 was approximately 25 per 100,000 population. In a city of 40,000 what is the probability of having 5 or fewer cases? In a city of
Answer
3.2. A manufacturer of wool blankets inspects the blankets by counting the number of defects. (A defect may be a tear, an oil spot, etc.) From past records it is known that the mean number of defects per blanket is 5. Calculate the probability that a blanket will contain 2 or more defects.
3.3. Bank tellers in a certain bank make errors in entering figures in their ledgers at the rate of 0.75 error per page of entries. What is the probability that in 4 pages there will be 2 or more errors?
Answer
0.8008.
3.4. Workers in a certain factory incur accidents at the rate of 2 accidents per week. Calculate the probability that there will be 2 or fewer accidents during (i) 1 week, (ii) 2 weeks; (iii) calculate the probability that there will be 2 or fewer accidents in each of 2 weeks.
3.5. A radioactive source is observed during 4 time intervals of 6 seconds each. The number of particles emitted during each period are counted. If the particles emitted obey a Poisson probability law, at a rate of 0.5 particles emitted per second, find the probability that (i) in each of the 4 time intervals 3 or more particles will be emitted, (ii) in at least 1 of the 4 time intervals 3 or more particles will be emitted.
Answer
(i) 0.111; (ii) 0.968.
3.6. Suppose that the suicide rate in a certain state is 1 suicide per 250,000 inhabitants per week.
(i) Find the probability that in a certain town of population 500,000 there will be 6 or more suicides in a week.
(ii) What is the expected number of weeks in a year in which 6 or more suicides will be reported in this town.
(iii) Would you find it surprising that during 1 year there were at least 2 weeks in which 6 or more suicides were reported?
3.7. Suppose that customers enter a certain shop at the rate of 30 persons an hour.
(i) What is the probability that during a 2-minute interval either no one will enter the shop or at least 2 persons will enter the shop.
(ii) If you observed the number of persons entering the shop during each of 302 -minute intervals, would you find it surprising that 20 or more of these intervals had the property that either no one or at least 2 persons entered the shop during that time?
Answer
(i) 0.632; (ii) not surprising, since the number of 2 minute intervals in an hour in which either no one enters or 2 or more enter obeys a binomial probability law with mean 19.0 and variance 6.975.
3.8. Suppose that the telephone calls coming into a certain switchboard obey a Poisson probability law at a rate of 16 calls per minute. If the switchboard can handle at most 24 calls per minute, what is the probability, using a normal approximation, that in 1 minute the switchboard will receive more calls than it can handle (assume all lines are clear).
3.9. In a large fleet of delivery trucks the average number inoperative on any day because of repairs is 2. Two standby trucks are available. What is the probability that on any day (i) no standby trucks will be needed, (ii) the number of standby trucks is inadequate.
Answer
(i) 0.1353; (ii) 0.3233.
3.10. Major motor failures occur among the buses of a large bus company at the rate of 2 a day. Assuming that each motor failure requires the services of 1 mechanic for a whole day, how many mechanics should the bus company employ to insure that the probability is at least 0.95 that a mechanic will be available to repair each motor as it fails? (More precisely, find the smallest integer
3.11. Consider a restaurant located in the business section of a city. How many seats should it have available if it wishes to serve at least
(i) 1000 persons pass by the restaurant in a given hour, each of whom has probability
(ii) persons, each of whom has probability
(iii) persons, desiring to be patrons of the restaurant, arrive at the restaurant at the rate of 10 an hour.
Answer
15.
3.12. Flying-bomb hits on London. The following data (R. D. Clarke, “An application of the Poisson distribution”, Journal of the Institute of Actuaries , Vol. 72 (1946), p. 48) give the number of flying-bomb hits recorded in each of 576 small areas of
Using the procedure in example
3.13. For each of the following numerical valued random phenomena state conditions under which it may be expected to obey, either exactly or approximately, a Poisson probability law: (i) the number of telephone calls received at a given switchboard per minute; (ii) the number of automobiles passing a given point on a highway per minute; (iii) the number of bacterial colonies in a given culture per 0.01 square millimeter on a microscope slide; (iv) the number of times one receives 4 aces per 75 hands of bridge; (v) the number of defective screws per box of 100.
