Poisson Distribution and the Poisson Process

5.5 Poisson Distribution and the Poisson Process

Experiments yielding numerical values of a random variable X, the number of outcomes occurring during a given time interval or in a specified region, are called Poisson experiments . The given time interval may be of any length, such as a minute, a day, a week, a month, or even a year. For example, a Poisson experiment can generate observations for the random variable X representing the number of telephone calls received per hour by an office, the number of days school is closed due to snow during the winter, or the number of games postponed due to rain during a baseball season. The specified region could be a line segment, an area,

a volume, or perhaps a piece of material. In such instances, X might represent the number of field mice per acre, the number of bacteria in a given culture, or the number of typing errors per page. A Poisson experiment is derived from the Poisson process and possesses the following properties.

Properties of the Poisson Process

1. The number of outcomes occurring in one time interval or specified region of space is independent of the number that occur in any other disjoint time in- terval or region. In this sense we say that the Poisson process has no memory.

2. The probability that a single outcome will occur during a very short time interval or in a small region is proportional to the length of the time interval or the size of the region and does not depend on the number of outcomes occurring outside this time interval or region.

3. The probability that more than one outcome will occur in such a short time interval or fall in such a small region is negligible.

The number X of outcomes occurring during a Poisson experiment is called a Poisson random variable , and its probability distribution is called the Poisson

162 Chapter 5 Some Discrete Probability Distributions

distribution . The mean number of outcomes is computed from µ = λt, where t is the specific “time,” “distance,” “area,” or “volume” of interest. Since the

probabilities depend on λ, the rate of occurrence of outcomes, we shall denote them by p(x; λt). The derivation of the formula for p(x; λt), based on the three properties of a Poisson process listed above, is beyond the scope of this book. The following formula is used for computing Poisson probabilities.

Poisson The probability distribution of the Poisson random variable X, representing Distribution the number of outcomes occurring in a given time interval or specified region denoted by t, is

e − λt (λt) x

p(x; λt) =

, x = 0, 1, 2, . . . ,

x!

where λ is the average number of outcomes per unit time, distance, area, or volume and e = 2.71828 . . . .

Table A.2 contains Poisson probability sums,

P (r; λt) =

p(x; λt),

x=0

for selected values of λt ranging from 0.1 to 18.0. We illustrate the use of this table with the following two examples.

Example 5.17: During a laboratory experiment, the average number of radioactive particles pass- ing through a counter in 1 millisecond is 4. What is the probability that 6 particles enter the counter in a given millisecond?

Solution : Using the Poisson distribution with x = 6 and λt = 4 and referring to Table A.2, we have

p(6; 4) =

p(x; 4) −

p(x; 4) = 0.8893 − 0.7851 = 0.1042.

Example 5.18: Ten is the average number of oil tankers arriving each day at a certain port. The facilities at the port can handle at most 15 tankers per day. What is the probability

that on a given day tankers have to be turned away? Solution : Let X be the number of tankers arriving each day. Then, using Table A.2, we have

P (X > 15) = 1 − P (X ≤ 15) = 1 −

p(x; 10) = 1 − 0.9513 = 0.0487.

x=0

Like the binomial distribution, the Poisson distribution is used for quality con- trol, quality assurance, and acceptance sampling. In addition, certain important continuous distributions used in reliability theory and queuing theory depend on the Poisson process. Some of these distributions are discussed and developed in Chapter 6. The following theorem concerning the Poisson random variable is given in Appendix A.25.

Theorem 5.4: Both the mean and the variance of the Poisson distribution p(x; λt) are λt.

5.5 Poisson Distribution and the Poisson Process 163

Nature of the Poisson Probability Function

Like so many discrete and continuous distributions, the form of the Poisson distri- bution becomes more and more symmetric, even bell-shaped, as the mean grows large. Figure 5.1 illustrates this, showing plots of the probability function for µ = 0.1, µ = 2, and µ = 5. Note the nearness to symmetry when µ becomes as large as 5. A similar condition exists for the binomial distribution, as will be illustrated later in the text.

Figure 5.1: Poisson density functions for different means.

Approximation of Binomial Distribution by a Poisson Distribution

It should be evident from the three principles of the Poisson process that the Poisson distribution is related to the binomial distribution. Although the Poisson usually finds applications in space and time problems, as illustrated by Examples

5.17 and 5.18, it can be viewed as a limiting form of the binomial distribution. In the case of the binomial, if n is quite large and p is small, the conditions begin to simulate the continuous space or time implications of the Poisson process. The in- dependence among Bernoulli trials in the binomial case is consistent with principle

2 of the Poisson process. Allowing the parameter p to be close to 0 relates to prin- ciple 3 of the Poisson process. Indeed, if n is large and p is close to 0, the Poisson distribution can be used, with µ = np, to approximate binomial probabilities. If p is close to 1, we can still use the Poisson distribution to approximate binomial probabilities by interchanging what we have defined to be a success and a failure, thereby changing p to a value close to 0.

Theorem 5.5: Let X be a binomial random variable with probability distribution b(x; n, p). When n → ∞, p → 0, and np n→∞ −→ µ remains constant,

b(x; n, p) n→∞ −→ p(x; µ).

164 Chapter 5 Some Discrete Probability Distributions

Example 5.19: In a certain industrial facility, accidents occur infrequently. It is known that the probability of an accident on any given day is 0.005 and accidents are independent of each other.

(a) What is the probability that in any given period of 400 days there will be an

accident on one day? (b) What is the probability that there are at most three days with an accident? Solution : Let X be a binomial random variable with n = 400 and p = 0.005. Thus, np = 2.

Using the Poisson approximation,

− 2 (a) P (X = 1) = e 1 2 = 0.271 and

− (b) P (X ≤ 3) = 2 e 2 x /x! = 0.857.

x=0

Example 5.20: In a manufacturing process where glass products are made, defects or bubbles occur, occasionally rendering the piece undesirable for marketing. It is known that, on average, 1 in every 1000 of these items produced has one or more bubbles. What is the probability that a random sample of 8000 will yield fewer than 7 items possessing bubbles?

Solution : This is essentially a binomial experiment with n = 8000 and p = 0.001. Since p is very close to 0 and n is quite large, we shall approximate with the Poisson distribution using

Hence, if X represents the number of bubbles, we have

P (X < 7) =

b(x; 8000, 0.001) ≈ p(x; 8) = 0.3134.

x=0

Exercises

5.49 The probability that a person living in a certain disease. If the probability of contracting the disease is city owns a dog is estimated to be 0.3. Find the prob- 1/6, what is the probability that 8 mice are required? ability that the tenth person randomly interviewed in that city is the fifth one to own a dog.

5.53 An inventory study determines that, on aver- age, demands for a particular item at a warehouse are 5.50 Find the probability that a person flipping a coin made 5 times per day. What is the probability that on gets

a given day this item is requested (a) the third head on the seventh flip;

(a) more than 5 times?

(b) the first head on the fourth flip.

(b) not at all?

5.51 Three people toss a fair coin and the odd one 5.54 According to a study published by a group of pays for coffee. If the coins all turn up the same, they University of Massachusetts sociologists, about two- are tossed again. Find the probability that fewer than thirds of the 20 million persons in this country who

4 tosses are needed. take Valium are women. Assuming this figure to be a valid estimate, find the probability that on a given day

5.52 A scientist inoculates mice, one at a time, with the fifth prescription written by a doctor for Valium is a disease germ until he finds 2 that have contracted the (a) the first prescribing Valium for a woman;

Exercises 165 (b) the third prescribing Valium for a woman.

find the probability that (a) fewer than 5 fail the test;

5.55 The probability that a student pilot passes the (b) 8, 9, or 10 fail the test. written test for a private pilot’s license is 0.7. Find the probability that a given student will pass the test

5.63 Find the mean and variance of the random vari- (a) on the third try;

able X in Exercise 5.58, representing the number of (b) before the fourth try.

hurricanes per year to hit a certain area of the eastern United States.

5.56 On average, 3 traffic accidents per month occur 5.64 Find the mean and variance of the random vari- at a certain intersection. What is the probability that able X in Exercise 5.61, representing the number of in any given month at this intersection

persons among 10,000 who make an error in preparing (a) exactly 5 accidents will occur?

their income tax returns.

(b) fewer than 3 accidents will occur? 5.65 An automobile manufacturer is concerned about (c) at least 2 accidents will occur?

a fault in the braking mechanism of a particular model. The fault can, on rare occasions, cause a catastrophe at 5.57 On average, a textbook author makes two word- high speed. The distribution of the number of cars per processing errors per page on the first draft of her text- year that will experience the catastrophe is a Poisson book. What is the probability that on the next page random variable with λ = 5. she will make

(a) What is the probability that at most 3 cars per year (a) 4 or more errors?

will experience a catastrophe? (b) no errors?

(b) What is the probability that more than 1 car per year will experience a catastrophe?

5.58 A certain area of the eastern United States is, 5.66 Changes in airport procedures require consid- on average, hit by 6 hurricanes a year. Find the prob- erable planning. Arrival rates of aircraft are impor- ability that in a given year that area will be hit by

tant factors that must be taken into account. Suppose (a) fewer than 4 hurricanes;

small aircraft arrive at a certain airport, according to (b) anywhere from 6 to 8 hurricanes.

a Poisson process, at the rate of 6 per hour. Thus, the Poisson parameter for arrivals over a period of hours is

5.59 Suppose the probability that any given person µ = 6t. will believe a tale about the transgressions of a famous (a) What is the probability that exactly 4 small air-

actress is 0.8. What is the probability that craft arrive during a 1-hour period? (a) the sixth person to hear this tale is the fourth one (b) What is the probability that at least 4 arrive during

to believe it?

a 1-hour period?

(b) the third person to hear this tale is the first one to (c) If we define a working day as 12 hours, what is believe it?

the probability that at least 75 small aircraft ar- rive during a working day?

5.60 The average number of field mice per acre in a 5-acre wheat field is estimated to be 12. Find the 5.67 The number of customers arriving per hour at a

probability that fewer than 7 field mice are found certain automobile service facility is assumed to follow a Poisson distribution with mean λ = 7.

(a) on a given acre; (a) Compute the probability that more than 10 cus- (b) on 2 of the next 3 acres inspected.

tomers will arrive in a 2-hour period. (b) What is the mean number of arrivals during a

5.61 Suppose that, on average, 1 person in 1000

2-hour period?

makes a numerical error in preparing his or her income tax return. If 10,000 returns are selected at random 5.68 Consider Exercise 5.62. What is the mean num- and examined, find the probability that 6, 7, or 8 of ber of students who fail the test? them contain an error.

5.69 The probability that a person will die when he 5.62 The probability that a student at a local high or she contracts a virus infection is 0.001. Of the next

school fails the screening test for scoliosis (curvature 4000 people infected, what is the mean number who of the spine) is known to be 0.004. Of the next 1875 will die? students at the school who are screened for scoliosis,

166 Chapter 5 Some Discrete Probability Distributions

5.70 A company purchases large lots of a certain kind modate the patient traffic if there are more than 10 of electronic device. A method is used that rejects a emergency cases in a given hour. It is assumed that lot if 2 or more defective units are found in a random patient arrival follows a Poisson process, and historical sample of 100 units.

data suggest that, on the average, 5 emergencies arrive (a) What is the mean number of defective units found per hour. in a sample of 100 units if the lot is 1% defective?

(a) What is the probability that in a given hour the (b) What is the variance?

staff cannot accommodate the patient traffic? (b) What is the probability that more than 20 emer- 5.71 For a certain type of copper wire, it is known

gencies arrive during a 3-hour shift? that, on the average, 1.5 flaws occur per millimeter.

Assuming that the number of flaws is a Poisson random 5.74 It is known that 3% of people whose luggage variable, what is the probability that no flaws occur in is screened at an airport have questionable objects in a certain portion of wire of length 5 millimeters? What their luggage. What is the probability that a string of is the mean number of flaws in a portion of length 5

15 people pass through screening successfully before an millimeters?

individual is caught with a questionable object? What is the expected number of people to pass through be-

5.72 Potholes on a highway can be a serious problem, fore an individual is stopped? and are in constant need of repair. With a particular

type of terrain and make of concrete, past experience 5.75 Computer technology has produced an environ- suggests that there are, on the average, 2 potholes per ment in which robots operate with the use of micro- mile after a certain amount of usage. It is assumed processors. The probability that a robot fails during that the Poisson process applies to the random vari- any 6-hour shift is 0.10. What is the probability that able “number of potholes.”

a robot will operate through at most 5 shifts before it (a) What is the probability that no more than one pot- fails? hole will appear in a section of 1 mile?

(b) What is the probability that no more than 4 pot- 5.76 The refusal rate for telephone polls is known to holes will occur in a given section of 5 miles?

be approximately 20%. A newspaper report indicates that 50 people were interviewed before the first refusal.

5.73 Hospital administrators in large cities anguish (a) Comment on the validity of the report. Use a prob- about traffic in emergency rooms. At a particular hos-

ability in your argument.

pital in a large city, the staff on hand cannot accom- (b) What is the expected number of people interviewed

before a refusal?

Review Exercises

5.77 During a manufacturing process, 15 units are welds. Otherwise, it will not be considered efficient. randomly selected each day from the production line

A test is to be conducted with a prototype that is to to check the percent defective. From historical infor- perform 100 welds. The machine will be accepted for mation it is known that the probability of a defective manufacture if it misses no more than 3 welds. unit is 0.05. Any time 2 or more defectives are found (a) What is the probability that a good machine will in the sample of 15, the process is stopped. This proce-

be rejected?

dure is used to provide a signal in case the probability of a defective has increased.

(b) What is the probability that an inefficient machine with 95% welding success will be accepted?

(a) What is the probability that on any given day the production process will be stopped? (Assume 5% defective.)

5.79 A car rental agency at a local airport has avail- able 5 Fords, 7 Chevrolets, 4 Dodges, 3 Hondas, and 4

(b) Suppose that the probability of a defective has in- Toyotas. If the agency randomly selects 9 of these cars creased to 0.07. What is the probability that on to chauffeur delegates from the airport to the down- any given day the production process will not be town convention center, find the probability that 2 stopped?

Fords, 3 Chevrolets, 1 Dodge, 1 Honda, and 2 Toyotas are used.

5.78 An automatic welding machine is being consid- ered for use in a production process. It will be con- 5.80 Service calls come to a maintenance center ac- sidered for purchase if it is successful on 99% of its cording to a Poisson process, and on average, 2.7 calls

Review Exercises 167 are received per minute. Find the probability that

will win back at least the cost of 3 tickets. (a) no more than 4 calls come in any minute;

5.87 Imperfections in computer circuit boards and (b) fewer than 2 calls come in any minute;

computer chips lend themselves to statistical treat- (c) more than 10 calls come in a 5-minute period.

ment. For a particular type of board, the probability of a diode failure is 0.03 and the board contains 200

5.81 An electronics firm claims that the proportion of diodes. defective units from a certain process is 5%. A buyer (a) What is the mean number of failures among the has a standard procedure of inspecting 15 units selected

diodes?

randomly from a large lot. On a particular occasion, (b) What is the variance? the buyer found 5 items defective.

(c) The board will work if there are no defective diodes. (a) What is the probability of this occurrence, given

What is the probability that a board will work? that the claim of 5% defective is correct?

(b) What would be your reaction if you were the buyer? 5.88 The potential buyer of a particular engine re- quires (among other things) that the engine start suc-

5.82 An electronic switching device occasionally mal- cessfully 10 consecutive times. Suppose the probability functions, but the device is considered satisfactory if it of a successful start is 0.990. Let us assume that the makes, on average, no more than 0.20 error per hour. outcomes of attempted starts are independent.

A particular 5-hour period is chosen for testing the de- (a) What is the probability that the engine is accepted vice. If no more than 1 error occurs during the time

after only 10 starts?

period, the device will be considered satisfactory. (b) What is the probability that 12 attempted starts (a) What is the probability that a satisfactory device

are made during the acceptance process? will be considered unsatisfactory on the basis of the

test? Assume a Poisson process. 5.89 The acceptance scheme for purchasing lots con- (b) What is the probability that a device will be ac- taining a large number of batteries is to test no more

cepted as satisfactory when, in fact, the mean num- than 75 randomly selected batteries and to reject a lot ber of errors is 0.25? Again, assume a Poisson pro- if a single battery fails. Suppose the probability of a cess.

failure is 0.001. (a) What is the probability that a lot is accepted?

5.83 A company generally purchases large lots of a certain kind of electronic device. A method is used (b) What is the probability that a lot is rejected on the that rejects a lot if 2 or more defective units are found

20th test?

in a random sample of 100 units. (c) What is the probability that it is rejected in 10 or (a) What is the probability of rejecting a lot that is 1%

fewer trials?

defective? (b) What is the probability of accepting a lot that is 5.90 An oil drilling company ventures into various lo-

5% defective? cations, and its success or failure is independent from one location to another. Suppose the probability of a success at any specific location is 0.25.

5.84 A local drugstore owner knows that, on average, 100 people enter his store each hour.

(a) What is the probability that the driller drills at 10 locations and has 1 success?

(a) Find the probability that in a given 3-minute pe- riod nobody enters the store.

(b) The driller will go bankrupt if it drills 10 times be- fore the first success occurs. What are the driller’s

(b) Find the probability that in a given 3-minute pe- prospects for bankruptcy? riod more than 5 people enter the store.

5.91 Consider the information in Review Exercise 5.85 (a) Suppose that you throw 4 dice. Find the

5.90. The drilling company feels that it will “hit it probability that you get at least one 1.

big” if the second success occurs on or before the sixth (b) Suppose that you throw 2 dice 24 times. Find the attempt. What is the probability that the driller will probability that you get at least one (1, 1), that is, hit it big? “snake-eyes.”

5.92 A couple decides to continue to have children un- 5.86 Suppose that out of 500 lottery tickets sold, 200 til they have two males. Assuming that P (male) = 0.5, pay off at least the cost of the ticket. Now suppose what is the probability that their second male is their that you buy 5 tickets. Find the probability that you fourth child?

168 Chapter 5 Some Discrete Probability Distributions

5.93 It is known by researchers that 1 in 100 people system is unworthy and must be improved. carries a gene that leads to the inheritance of a certain (a) What is the probability that an incoming missile chronic disease. In a random sample of 1000 individ-

will not be detected by any of the three screens? uals, what is the probability that fewer than 7 indi-

viduals carry the gene? Use a Poisson approximation. (b) What is the probability that the missile will be de- Again, using the approximation, what is the approxi-

tected by only one screen? mate mean number of people out of 1000 carrying the (c) What is the probability that it will be detected by

gene? at least two out of three screens? 5.94 A production process produces electronic com- 5.98 Suppose it is important that the overall missile

ponent parts. It is presumed that the probability of a defense system be as near perfect as possible. defective part is 0.01. During a test of this presump- (a) Assuming the quality of the screens is as indicated tion, 500 parts are sampled randomly and 15 defectives

in Review Exercise 5.97, how many are needed are observed.

to ensure that the probability that a missile gets (a) What is your response to the presumption that the

through undetected is 0.0001? process is 1% defective? Be sure that a computed (b) Suppose it is decided to stay with only 3 screens

probability accompanies your comment. and attempt to improve the screen detection abil- (b) Under the presumption of a 1% defective process,

ity. What must the individual screen effectiveness what is the probability that only 3 parts will be

(i.e., probability of detection) be in order to achieve found defective?

the effectiveness required in part (a)? (c) Do parts (a) and (b) again using the Poisson ap-

proximation. 5.99 Go back to Review Exercise 5.95(a). Re- compute the probability using the binomial distribu- 5.95 A production process outputs items in lots of 50. tion. Comment. Sampling plans exist in which lots are pulled aside pe- riodically and exposed to a certain type of inspection. 5.100 There are two vacancies in a certain university It is usually assumed that the proportion defective is statistics department. Five individuals apply. Two very small. It is important to the company that lots have expertise in linear models, and one has exper- containing defectives be a rare event. The current in- tise in applied probability. The search committee is spection plan is to periodically sample randomly 10 out instructed to choose the two applicants randomly. of the 50 items in a lot and, if none are defective, to (a) What is the probability that the two chosen are perform no intervention.

those with expertise in linear models? (a) Suppose in a lot chosen at random, 2 out of 50 are (b) What is the probability that of the two chosen, one

defective. What is the probability that at least 1 has expertise in linear models and one has expertise in the sample of 10 from the lot is defective?

in applied probability?

(b) From your answer to part (a), comment on the quality of this sampling plan.

5.101 The manufacturer of a tricycle for children has (c) What is the mean number of defects found out of received complaints about defective brakes in the prod- 10 items sampled?

uct. According to the design of the product and consid- erable preliminary testing, it had been determined that the probability of the kind of defect in the complaint

5.96 Consider the situation of Review Exercise 5.95. was 1 in 10,000 (i.e., 0.0001). After a thorough investi- It has been determined that the sampling plan should gation of the complaints, it was determined that during be extensive enough that there is a high probability,

a certain period of time, 200 products were randomly say 0.9, that if as many as 2 defectives exist in the lot chosen from production and 5 had defective brakes.

of 50 being sampled, at least 1 will be found in the sampling. With these restrictions, how many of the 50 (a) Comment on the “1 in 10,000” claim by the man- items should be sampled?

ufacturer. Use a probabilistic argument. Use the binomial distribution for your calculations.

5.97 National security requires that defense technol- (b) Repeat part (a) using the Poisson approximation. ogy be able to detect incoming projectiles or missiles. To make the defense system successful, multiple radar 5.102 Group Project : Divide the class into two screens are required. Suppose that three independent groups of approximately equal size. The students in

screens are to be operated and the probability that any group 1 will each toss a coin 10 times (n 1 ) and count one screen will detect an incoming missile is 0.8. Ob- the number of heads obtained. The students in group 2 viously, if no screens detect an incoming projectile, the will each toss a coin 40 times (n 2 ) and again count the

5.6 Potential Misconceptions and Hazards 169 number of heads. The students in each group should (b) Referring again to Theorem 5.1, show that

individually compute the proportion of heads observed, which is an estimate of p, the probability of observing

E (p 1 ) = E(p 2 ) = p = 0.5.

a head. Thus, there will be a set of values of p 1 (from

group 1) and a set of values p 2 (from group 2). All of

1 = n 1 is 4 times the value of the true value of the probability of observing a head

the values of p 1 and p 2 are estimates of 0.5, which is

2 (c) Show that σ X

for a fair coin.

. Then explain further why the values values of p 1 or p 2 ? Consider the proof of Theorem

(a) Which set of values is consistently closer to 0.5, the

of p 2 from group 2 are more consistently closer to 5.1 on page 147 with regard to the estimates of the

the true value, p = 0.5, than the values of p 1 from

parameter p = 0.5. The values of p 1 were obtained

group 1.

with n = n 1 = 10, and the values of p 2 were ob- You will continue to learn more and more about tained with n = n 2 = 40. Using the notation of the proof, the estimates are given by

parameter estimation beginning in Chapter 9. At that point emphasis will put on the importance of

1 x 1 I 1 p n +···+I = = 1

the mean and variance of an estimator of a param- eter.

where I 1 ,...,I n 1 are 0s and 1s and n 1 = 10, and

x 2 p 1 2 I +···+I n

n 2 n 2 where I 1 ,...,I n 2 , again, are 0s and 1s and n 2 = 40.