(the Poisson model is suggested in the article “Analysis of Random Telegraph Noise in 45-nm CMOS Using On-Chip Characterization System ”(IEEE Trans.

5 2 (the Poisson model is suggested in the article “Analysis of Random Telegraph Noise in 45-nm CMOS Using On-Chip Characterization System ”(IEEE Trans.

  on Electron Devices, 2013: 1716–1722); we changed the value of the parameter for computational ease).

  The probability that there are exactly three traps is

  e 2 3

  P (X 5 3) 5 p(3;2) 5

  3! and the probability that there are at most three traps is

  x5 o 0 !

  3 e 2 x

  P (X 3) 5 F(3; 2) 5

  This latter cumulative probability is found at the intersection of the μ 5 2 column and the x 5 3 row of Appendix Table A.2, whereas p(3;2) 5 F(3;2) 2 F(2;2) 5 .857 2 .677 5 .180, the difference between two consecutive entries in the m 5 2 column of the cumulative Poisson table.

  n

  the Poisson distribution as a Limit

  The rationale for using the Poisson distribution in many situations is provided by the following proposition.

  pROpOSITION

  Suppose that in the binomial pmf b(x; n, p), we let n S ` and p S 0 in such

  a way that np approaches a value m . 0. Then b(x; n, p) S p(x; m).

  According to this result, in any binomial experiment in which n is large and p is small, b(x; n, p) < p(x; m), where m 5 np. As a rule of thumb, this approximation can safely be applied if n . 50 and np , 5.

  3.6 the poisson probability Distribution 133

  ExamplE 3.39 If a publisher of nontechnical books takes great pains to ensure that its books are free of typographical errors, so that the probability of any given page containing at least one such error is .005 and errors are independent from page to page, what is the probability that one of its 600-page novels will contain exactly one page with errors? At most three pages with errors?

  With S denoting a page containing at least one error and F an error-free page, the number X of pages containing at least one error is a binomial rv with n 5 600 and p 5 .005, so np 5 3. We wish

  e 2 3 (3) 1

  P (X 5 1) 5 b(1; 600, .005) < p(1; 3) 5

  The binomial value is b(1; 600, .005) 5 .14899, so the approximation is very good.

  x5 o 0

  p (x; 3) 5 F(3;3) 5 .647

  which to three-decimal-place accuracy is identical to B(3; 600, .005).

  n

  Table 3.2 shows the Poisson distribution for m 5 3 along with three bino- mial distributions with np 5 3, and Figure 3.8 plots the Poisson along with the first two binomial distributions. The approximation is of limited use for n 5 30, but of course the accuracy is better for n 5 100 and much better for n 5 300.

  Table 3.2 Comparing the Poisson and Three Binomial Distributions

  x

  n 5 30, p 5 .1

  n 5 100, p 5 .03

  n 5 300, p 5 .01 Poisson, m53

  Bin, n530 ( o); Bin, n5100 (x); Poisson ( )

  Figure 3.8 Comparing a Poisson and two binomial distributions

  134 Chapter 3 Discrete random Variables and probability Distributions

  the Mean and Variance of X

  Since b(x; n, p) S p(x; m) as n S `, p S 0, np S m, the mean and variance of

  a binomial variable should approach those of a Poisson variable. These limits are np S m and np(1 2 p) S m.

  pROpOSITION

  If X has a Poisson distribution with parameter m, then E(X) 5 V(X) 5 m.

  These results can also be derived directly from the definitions of mean and variance.

  ExamplE 3.40

  Both the expected number of traps and the variance of the number of traps equal 2,

  (Example 3.38

  and s X 5 Ïm 5 Ï2 5 1.414.

  n

  continued)

  the Poisson Process

  A very important application of the Poisson distribution arises in connection with the occurrence of events of some type over time. Events of interest might be visits to a particular Web site, pulses of some sort recorded by a counter, email messages sent to a particular address, accidents in an industrial facility, or cosmic ray showers observed by astronomers at a particular observatory. We make the following assump- tions about the way in which the events of interest occur:

  1. There exists a parameter a . 0 such that for any short time interval of length

  D t , the probability that exactly one event occurs is a ? Dt 1 o(Dt).

  2. The probability of more than one event occurring during Dt is o(Dt) [which, along with Assumption 1, implies that the probability of no events during Dt is

  1 2 a ? Dt 2 o(Dt)].

  3. The number of events occurring during the time interval Dt is independent of the number that occur prior to this time interval.

  Informally, Assumption 1 says that for a short interval of time, the probability of a single event occurring is approximately proportional to the length of the time inter- val, where a is the constant of proportionality. Now let Pk(t) denote the probability that k events will be observed during any particular time interval of length t.

  pROpOSITION

  P (t) 5 e 2 at ? (at) k k yk!, so that the number of events during a time interval of length t is a Poisson rv with parameter m 5 at. The expected number of events during any such time interval is then at, so the expected number during a unit interval of time is a.

  The occurrence of events over time as described is called a Poisson process; the parameter a specifies the rate for the process.

  ExamplE 3.41 Suppose pulses arrive at a counter at an average rate of six per minute, so that a 5 6. To find the probability that in a .5-min interval at least one pulse is received, note that the number of pulses in such an interval has a Poisson distribution with parameter

  A quantity is o(Dt) (read “little o of delta t”) if, as Dt approaches 0, so does o(Dt)Dt. That is, o(Dt) is even

  more negligible (approaches 0 faster) than Dt itself. The quantity (Dt) 2 has this property, but sin(Dt) does not.

  3.6 the poisson probability Distribution 135

  a t5 6(.5) 5 3 (.5 min is used because a is expressed as a rate per minute). Then with

  X5 the number of pulses received in the 30-sec interval,

  Instead of observing events over time, consider observing events of some type that occur in a two- or three-dimensional region. For example, we might select on a map a certain region R of a forest, go to that region, and count the number of trees. Each tree would represent an event occurring at a particular point in space. Under assumptions similar to 1–3, it can be shown that the number of events occurring in a region R has a Poisson distribution with parameter a ? a(R), where a(R) is the area of R. The quantity a is the expected number of events per unit area or volume.

  ExERcisEs Section 3.6 (79–93)

  79. The article “Expectation Analysis of the Probability of

  c. Be between 10 and 20, inclusive? Be strictly between

  Failure for Water Supply Pipes” (J. of Pipeline

  10 and 20?

  Systems Engr. and Practice, May 2012: 36–46) pro-

  d. Be within 2 standard deviations of the mean value?

  posed using the Poisson distribution to model the num-

  82. Consider writing onto a computer disk and then sending

  ber of failures in pipelines of various types. Suppose that

  it through a certifier that counts the number of missing

  for cast-iron pipe of a particular length, the expected

  pulses. Suppose this number X has a Poisson distribu-

  number of failures is 1 (very close to one of the cases

  tion with parameter m 5 .2. (Suggested in “Average

  considered in the article). Then X, the number of failures,

  Sample Number for Semi-Curtailed Sampling Using

  has a Poisson distribution with m 5 1.

  the Poisson Distribution,” J. Quality Technology,

  a. Obtain P (X 5) by using Appendix Table A.2.

  1983: 126–129.)

  b. Determine P (X 5 2) first from the pmf formula and

  a. What is the probability that a disk has exactly one

  then from Appendix Table A.2.

  missing pulse?

  c. Determine P (2 X 4).

  b. What is the probability that a disk has at least two

  d. What is the probability that X exceeds its mean

  missing pulses?

  value by more than one standard deviation?

  c. If two disks are independently selected, what is the

  80. Let X be the number of material anomalies occurring in

  probability that neither contains a missing pulse?

  a particular region of an aircraft gas-turbine disk. The

  83. An article in the Los Angeles Times (Dec. 3, 1993)

  article “Methodology for Probabilistic Life Prediction

  reports that 1 in 200 people carry the defective gene that

  of Multiple-Anomaly Materials” (Amer. Inst. of

  causes inherited colon cancer. In a sample of 1000 indi-

  Aeronautics and Astronautics J., 2006: 787–793) pro-

  viduals, what is the approximate distribution of the num-

  poses a Poisson distribution for X. Suppose that m 5 4.

  ber who carry this gene? Use this distribution to calculate

  a. Compute both P(X 4) and P(X , 4).

  the approximate probability that

  b. Compute P(4 X 8).

  a. Between 5 and 8 (inclusive) carry the gene.

  c. Compute P (8 X).

  b. At least 8 carry the gene.

  d. What is the probability that the number of anomalies

  84. The Centers for Disease Control and Prevention

  exceeds its mean value by no more than one standard

  reported in 2012 that 1 in 88 American children had

  deviation?

  been diagnosed with an autism spectrum disorder

  81. Suppose that the number of drivers who travel between a

  (ASD).

  particular origin and destination during a designated time

  a. If a random sample of 200 American children is

  period has a Poisson distribution with parameter m 5 20

  selected, what are the expected value and standard

  (suggested in the article “Dynamic Ride Sharing: Theory

  deviation of the number who have been diagnosed with

  and Practice,” J. of Transp. Engr., 1997: 308–312). What

  ASD?

  is the probability that the number of drivers will

  b. Referring back to (a), calculate the approximate

  a. Be at most 10?

  probability that at least 2 children in the sample have

  b. Exceed 20?

  been diagnosed with ASD?

  136 Chapter 3 Discrete random Variables and probability Distributions

  c. If the sample size is 352, what is the approximate

  time. Suppose the mean time between occurrences of loads

  probability that fewer than 5 of the selected children

  is .5 year.

  have been diagnosed with ASD?

  a. How many loads can be expected to occur during a

  85. Suppose small aircraft arrive at a certain airport accord-

  2-year period?

  ing to a Poisson process with rate a 5 8 per hour, so that

  b. What is the probability that more than five loads

  the number of arrivals during a time period of t hours is

  occur during a 2-year period?

  a Poisson rv with parameter m 5 8t.

  c. How long must a time period be so that the probability

  a. What is the probability that exactly 6 small aircraft

  of no loads occurring during that period is at most .1?

  arrive during a 1-hour period? At least 6? At least 10?

  90. Let X have a Poisson distribution with parameter m.

  b. What are the expected value and standard deviation

  Show that E(X) 5 m directly from the definition of

  of the number of small aircraft that arrive during a

  expected value. [Hint: The first term in the sum equals 0,

  90-min period?

  and then x can be canceled. Now factor out m and show

  c. What is the probability that at least 20 small air-

  that what is left sums to 1.]

  craft arrive during a 2.5-hour period? That at most

  91. Suppose that trees are distributed in a forest according to

  10 arrive during this period?

  a two-dimensional Poisson process with parameter a, the

  86. Organisms are present in ballast water discharged from

  expected number of trees per acre, equal to 80.

  a ship according to a Poisson process with a concentra-

  a. What is the probability that in a certain quarter-acre

  tion of 10 organismsm 3 [the article “Counting at Low

  plot, there will be at most 16 trees?

  Concentrations: The Statistical Challenges of

  b. If the forest covers 85,000 acres, what is the expected

  Verifying Ballast Water Discharge Standards”

  number of trees in the forest?

  (Ecological Applications, 2013: 339–351) considers

  c. Suppose you select a point in the forest and construct

  using the Poisson process for this purpose].

  a circle of radius .1 mile. Let X 5 the number of

  a. What is the probability that one cubic meter of dis-

  trees within that circular region. What is the pmf of

  charge contains at least 8 organisms?

  X ? [Hint: 1 sq mile 5 640 acres.]

  b. What is the probability that the number of organisms

  92. Automobiles arrive at a vehicle equipment inspection sta-

  in 1.5 m 3

  of discharge exceeds its mean value by

  tion according to a Poisson process with rate a 5 10 per

  more than one standard deviation?

  hour. Suppose that with probability .5 an arriving vehicle

  c. For what amount of discharge would the probability

  will have no equipment violations.

  of containing at least 1 organism be .999?

  a. What is the probability that exactly ten arrive during

  87. The number of requests for assistance received by a tow-

  the hour and all ten have no violations?

  ing service is a Poisson process with rate a 5 4 per hour.

  b. For any fixed y 10, what is the probability that y

  a. Compute the probability that exactly ten requests are

  arrive during the hour, of which ten have no violations?

  received during a particular 2-hour period.

  c. What is the probability that ten “no-violation” cars

  b. If the operators of the towing service take a 30-min

  arrive during the next hour? [Hint: Sum the probabil-

  break for lunch, what is the probability that they do

  ities in part (b) from y 5 10 to `.]

  not miss any calls for assistance?

  93. a. In a Poisson process, what has to happen in both the

  c. How many calls would you expect during their

  time interval (0, t) and the interval (t, t 1 Dt) so that

  break?

  no events occur in the entire interval (0, t 1 Dt)? Use

  88. In proof testing of circuit boards, the probability that any

  this and Assumptions 1–3 to write a relationship

  particular diode will fail is .01. Suppose a circuit board

  between P 0 (t 1 Dt) and P 0 (t).

  contains 200 diodes.

  b. Use the result of part (a) to write an expression for

  a. How many diodes would you expect to fail, and what

  the difference P 0 (t 1 Dt) 2 P 0 (t). Then divide by Dt

  is the standard deviation of the number that are

  and let Dt S 0 to obtain an equation involving

  expected to fail?

  (ddt)P 0 (t), the derivative of P 0 (t) with respect to t. c. Verify that P (t) 5 e 2a b. What is the (approximate) probability that at least t 0 satisfies the equation of

  four diodes will fail on a randomly selected board?

  part (b).

  c. If five boards are shipped to a particular customer, how

  d. It can be shown in a manner similar to parts (a) and (b)

  likely is it that at least four of them will work prop-

  that the P k (t)s must satisfy the system of differential

  erly? (A board works properly only if all its diodes

  89. The article “Reliability-Based Service-Life Assessment

  dt

  of Aging Concrete Structures” (J. Structural Engr.,

  k 5 1, 2, 3,…

  1993: 1600–1621) suggests that a Poisson process can be

  Verify that P (t) 5 e 2a t (at) k yk! satisfies the system.

  used to represent the occurrence of structural loads over

  k

  (This is actually the only solution.)

  Supplementary exercises 137

  suPPlEmENTaRy ExERcisEs (94–122)

  94. Consider a deck consisting of seven cards, marked 1, 2,…,

  b. If the mean value of X is 2.313035, what is the prob-

  7. Three of these cards are selected at random. Define an

  ability that an individual wants at most 5 tattoos

  rv W by W 5 the sum of the resulting numbers, and com-

  removed?

  pute the pmf of W. Then compute m and s 2 . [Hint:

  c. Determine the standard deviation of X when the

  Consider outcomes as unordered, so that (1, 3, 7) and (3,

  mean value is as given in (b).

  1, 7) are not different outcomes. Then there are 35 out-

  [Note: The article “An Exploratory Investigation of

  comes, and they can be listed. (This type of rv actually

  Identity Negotiation and Tattoo Removal” (Academy

  arises in connection with a statistical procedure called

  of Marketing Science Review, vol. 12, no. 6, 2008) gave

  Wilcoxon’s rank-sum test, in which there is an x sample

  a sample of 22 observations on the number of tattoos

  and a y sample and W is the sum of the ranks of the x’s in

  people wanted removed; estimates of m and s calculated

  the combined sample; see Section 15.2.)

  from the data were 2.318182 and 1.249242, respectively.]

  95. After shuffling a deck of 52 cards, a dealer deals out 5.

  99. A k-out-of-n system is one that will function if and only

  Let X 5 the number of suits represented in the five-card

  if at least k of the n individual components in the system

  hand.

  function. If individual components function indepen-

  a. Show that the pmf of X is

  dently of one another, each with probability .9, what is

  x

  1 2 3 4 the probability that a 3-out-of-5 system functions?

  100. A manufacturer of integrated circuit chips wishes to con- trol the quality of its product by rejecting any batch in

  [Hint: p(1) 5 4P(all are spades), p(2) 5 6P(only spades

  which the proportion of defective chips is too high. To

  and hearts with at least one of each suit), and p(4)

  this end, out of each batch (10,000 chips), 25 will be

  5 4P(2 spades ù one of each other suit).]

  selected and tested. If at least 5 of these 25 are defective,

  b. Compute m, s 2 , and s.

  the entire batch will be rejected.

  96. The negative binomial rv X was defined as the number of

  a. What is the probability that a batch will be rejected

  F ’s preceding the rth S. Let Y 5 the number of trials

  if 5 of the chips in the batch are in fact defective?

  necessary to obtain the rth S. In the same manner in

  b. Answer the question posed in (a) if the percentage of

  which the pmf of X was derived, derive the pmf of Y.

  defective chips in the batch is 10.

  97. Of all customers purchasing automatic garage-door open-

  c. Answer the question posed in (a) if the percentage of

  ers, 75 purchase a chain-driven model. Let X 5 the

  defective chips in the batch is 20.

  number among the next 15 purchasers who select the

  d. What happens to the probabilities in (a)–(c) if the

  chain-driven model.

  critical rejection number is increased from 5 to 6?

  a. What is the pmf of X?

  101. Of the people passing through an airport metal detector,

  b. Compute P(X . 10).

  .5 activate it; let X 5 the number among a randomly

  c. Compute P (6 X 10).

  selected group of 500 who activate the detector.

  d. Compute m and s 2 .

  a. What is the (approximate) pmf of X?

  e. If the store currently has in stock 10 chain-driven

  b. Compute P(X 5 5).

  models and 8 shaft-driven models, what is the prob-

  c. Compute P (5 X).

  ability that the requests of these 15 customers can all

  102. An educational consulting firm is trying to decide

  be met from existing stock?

  whether high school students who have never before

  98. In some applications the distribution of a discrete rv X

  used a hand-held calculator can solve a certain type of

  resembles the Poisson distribution except that zero is not

  problem more easily with a calculator that uses reverse

  a possible value of X. For example, let X 5 the number

  Polish logic or one that does not use this logic. A sam-

  of tattoos that an individual wants removed when she or

  ple of 25 students is selected and allowed to practice on

  he arrives at a tattoo-removal facility. Suppose the pmf

  that a student worked the problem more quickly using

  both calculators. Then each student is asked to work one

  of X is

  problem on the reverse Polish calculator and a similar

  e 2u u x

  problem on the other. Let p 5 P(S), where S indicates

  a. Determine the value of k. Hint: The sum of all prob-

  reverse Polish logic than without, and let X 5 number

  abilities in the Poisson pmf is 1, and this pmf must

  of S’s.

  also sum to 1.

  a. If p 5 .5, what is P(7 X 18)? b. If p 5 .8, what is P(7 X 18)?

  138 Chapter 3 Discrete random Variables and probability Distributions

  c. If the claim that p 5 .5 is to be rejected when either

  a. Assuming the validity of this premise, among 25

  x 7 or x 18, what is the probability of rejecting

  randomly selected current customers, what is the

  the claim when it is actually correct?

  probability that between 2 and 6 (inclusive) qualify

  d. If the decision to reject the claim p 5 .5 is made as

  for membership?

  in part (c), what is the probability that the claim is

  b. Again assuming the validity of the premise, what are

  not rejected when p 5 .6? When p 5 .8?

  the expected number of customers who qualify and

  e. What decision rule would you choose for rejecting

  the standard deviation of the number who qualify in

  the claim p 5 .5 if you wanted the probability in part

  a random sample of 100 current customers?

  (c) to be at most .01?

  c. Let X denote the number in a random sample of 25

  103. Consider a disease whose presence can be identified by

  current customers who qualify for membership.

  carrying out a blood test. Let p denote the probability that

  Consider rejecting the company’s premise in favor of

  a randomly selected individual has the disease. Suppose

  the claim that p . .10 if x 7. What is the probabil-

  n individuals are independently selected for testing. One

  ity that the company’s premise is rejected when it is

  way to proceed is to carry out a separate test on each of

  actually valid?

  the n blood samples. A potentially more economical

  d. Refer to the decision rule introduced in part (c).

  approach, group testing, was introduced during World

  What is the probability that the company’s premise is

  War II to identify syphilitic men among army inductees.

  not rejected even though p 5 .20 (i.e., 20 qualify)?

  First, take a part of each blood sample, combine these

  107. Forty percent of seeds from maize (modern-day corn)

  specimens, and carry out a single test. If no one has the

  ears carry single spikelets, and the other 60 carry

  disease, the result will be negative, and only the one test

  paired spikelets. A seed with single spikelets will pro-

  is required. If at least one individual is diseased, the test

  duce an ear with single spikelets 29 of the time,

  on the combined sample will yield a positive result, in

  whereas a seed with paired spikelets will produce an ear

  which case the n individual tests are then carried out. If

  with single spikelets 26 of the time. Consider randomly

  p5 .1 and n 5 3, what is the expected number of tests

  selecting ten seeds.

  using this procedure? What is the expected number when

  a. What is the probability that exactly five of these

  n5 5? [The article “Random Multiple-Access

  seeds carry a single spikelet and produce an ear with

  Communication and Group Testing” (IEEE Trans. on

  a single spikelet?

  Commun., 1984: 769–774) applied these ideas to a com-

  b. What is the probability that exactly five of the ears

  munication system in which the dichotomy was active

  produced by these seeds have single spikelets? What

  idle user rather than diseasednondiseased.]

  is the probability that at most five ears have single

  104. Let p 1 denote the probability that any particular code

  spikelets?

  symbol is erroneously transmitted through a communica-

  108. A trial has just resulted in a hung jury because eight

  tion system. Assume that on different symbols, errors

  members of the jury were in favor of a guilty verdict and

  occur independently of one another. Suppose also that

  the other four were for acquittal. If the jurors leave the

  with probability p 2 an erroneous symbol is corrected

  jury room in random order and each of the first four

  upon receipt. Let X denote the number of correct symbols

  leaving the room is accosted by a reporter in quest of an

  in a message block consisting of n symbols (after the

  interview, what is the pmf of X 5 the number of jurors

  correction process has ended). What is the probability

  favoring acquittal among those interviewed? How many

  distribution of X?

  of those favoring acquittal do you expect to be inter-

  105. The purchaser of a power-generating unit requires c con-

  viewed?

  secutive successful start-ups before the unit will be

  109. A reservation service employs five information operators

  accepted. Assume that the outcomes of individual start-

  who receive requests for information independently of

  ups are independent of one another. Let p denote the

  one another, each according to a Poisson process with

  probability that any particular start-up is successful. The

  rate a 5 2 per minute.

  random variable of interest is X 5 the number of start-

  a. What is the probability that during a given 1-min

  ups that must be made prior to acceptance. Give the pmf

  period, the first operator receives no requests?

  of X for the case c 5 2. If p 5 .9, what is P(X 8)?

  b. What is the probability that during a given 1-min

  [Hint: For x 5, express p(x) “recursively” in terms of

  period, exactly four of the five operators receive no

  the pmf evaluated at the smaller values x 2 3, x 2 4, …, 2.]

  requests?

  (This problem was suggested by the article “Evaluation

  c. Write an expression for the probability that during a

  of a Start-Up Demonstration Test,” J. Quality

  given 1-min period, all of the operators receive

  Technology, 1983: 103–106.)

  exactly the same number of requests.

  106. A plan for an executive travelers’ club has been devel-

  110. Grasshoppers are distributed at random in a large field

  oped by an airline on the premise that 10 of its current

  according to a Poisson process with parameter a 5 2 per

  customers would qualify for membership.

  square yard. How large should the radius R of a circular

  Supplementary Exercises 139

  sampling region be taken so that the probability of find-

  b. What is the expected number of errors on the selected

  ing at least one in the region equals .99?

  form?

  111. A newsstand has ordered five copies of a certain issue of

  c. What is the variance of the number of errors on the

  a photography magazine. Let X 5 the number of individ-

  selected form?

  uals who come in to purchase this magazine. If X has a

  d. How does the pmf change if the first CPA prepares

  Poisson distribution with parameter m 5 4, what is the

  60 of all such forms and the second prepares 40?

  expected number of copies that are sold?

  116. The mode of a discrete random variable X with pmf p(x)

  112. Individuals A and B begin to play a sequence of chess

  is that value x for which p(x) is largest (the most proba-

  games. Let S 5 {A wins a game}, and suppose that out-

  ble x value).

  comes of successive games are independent with P(S) 5

  a. Let X , Bin(n, p). By considering the ratio

  p and P(F) 5 1 2 p (they never draw). They will play

  b (x 1 1; n,p) yb(x; n, p), show that b(x; n, p) increases

  until one of them wins ten games. Let X 5 the number of

  with x as long as x , np 2 (1 2 p). Conclude that

  games played (with possible values 10, 11,…, 19).

  the mode x is the integer satisfying (n 1 1)p2

  a. For x 5 10, 11, …, 19, obtain an expression for

  1 x (n 1 1)p.

  p (x) 5 P(X 5 x).

  b. Show that if X has a Poisson distribution with param-

  b. If a draw is possible, with p 5 P(S), q 5 P(F),

  eter m, the mode is the largest integer less than m. If

  1 2 p 2 q 5 P(draw), what are the possible values

  m is an integer, show that both m 2 1 and m are

  of X? What is P(20 X)? [Hint: P(20 X) 5

  modes.

  1 2 P(X , 20).]

  117. A computer disk storage device has ten concentric tracks,

  113. A test for the presence of a certain disease has probability

  numbered 1, 2,…, 10 from outermost to innermost, and a

  .20 of giving a false-positive reading (indicating that an

  single access arm. Let p i 5 the probability that any partic-

  individual has the disease when this is not the case) and

  ular request for data will take the arm to track

  probability .10 of giving a false-negative result. Suppose

  i (i 5 1,… , 10). Assume that the tracks accessed in succes-

  that ten individuals are tested, five of whom have the

  sive seeks are independent. Let X 5 the number of tracks

  disease and five of whom do not. Let X 5 the number of

  over which the access arm passes during two successive

  positive readings that result.

  requests (excluding the track that the arm has just left, so

  a. Does X have a binomial distribution? Explain your

  possible X values are x 5 0, 1, …, 9). Compute the pmf

  reasoning.

  of X. [Hint: P(the arm is now on track i and X 5 j) 5

  b. What is the probability that exactly three of the ten

  P (X 5 j|arm nowon i) ? p i . After the conditional

  test results are positive?

  pro bability is written in terms of p 1 ,…, p 10 , by the law of total probability, the desired probability is obtained by

  114. The generalized negative binomial pmf is given by

  summing over i.]

  nb(x; r, p) 5 k(r, x) ? p r (1 2 p) x

  118. If X is a hypergeometric rv, show directly from the defi-

  x 5 0, 1, 2,…

  nition that E(X) 5 nM yN (consider only the case n , M). [Hint: Factor nMN out of the sum for E(X), and show

  Let X, the number of plants of a certain species found in

  that the terms inside the sum are of the form

  a particular region, have this distribution with p 5 .3 and

  h (y; n 2 1, M 2 1, N 2 1), where y 5 x 2 1.]

  r5

  2.5. What is P(X 5 4)? What is the probability that at least one plant is found?

  119. Use the fact that

  (x 2 m) o 2 p (x)

  x : u x2muks o (x 2 m) 2 p (x)

  115. There are two Certified Public Accountants in a particu-

  lar office who prepare tax returns for clients. Suppose that for a particular type of complex form, the number of

  all x

  to prove Chebyshev’s inequality given in Exercise 44.

  errors made by the first preparer has a Poisson distribu-

  120. The simple Poisson process of Section 3.6 is character-

  tion with mean value m 1 , the number of errors made by

  ized by a constant rate a at which events occur per unit

  the second preparer has a Poisson distribution with mean

  time. A generalization of this is to suppose that the prob-

  value m 2 , and that each CPA prepares the same number of

  ability of exactly one event occurring in the interval

  forms of this type. Then if a form of this type is randomly

  [t, t 1 Dt] is a(t) ? Dt 1 o(Dt). It can then be shown that

  selected, the function

  the number of events occurring during an interval [t ,t ]

  has a Poisson distribution with parameter

  t

  t 1

  gives the pmf of X 5 the number of errors on the selected

  a. Verify that p(x; m 1 , m 2 ) is in fact a legitimate pmf

  The occurrence of events over time in this situation is

  ( 0 and sums to 1).

  called a nonhomogeneous Poisson process. The article “Inference Based on Retrospective Ascertainment,”

  140 Chapter 3 Discrete random Variables and probability Distributions

  (J. Amer. Stat. Assoc., 1989: 360–372), considers the

  duration of a data call to that same number is 1 minute.

  intensity function

  If 75 of all calls are voice calls, what is the expected

  duration of the next call?

  a (t) 5 e a1bt

  b. A deli sells three different types of chocolate chip

  as appropriate for events involving transmission of HIV

  purchasing a chocolate chip cookie select the first

  cookies. The number of chocolate chips in a type i

  (the AIDS virus) via blood transfusions. Suppose that

  cookie has a Poisson distribution with parameter

  a5 2 and b 5 .6 (close to values suggested in the paper),

  m i 5 i1 1 (i 5 1, 2, 3). If 20 of all customers

  with time in years.

  a. What is the expected number of events in the interval

  type, 50 choose the second type, and the remaining

  [0, 4]? In [2, 6]?

  30 opt for the third type, what is the expected num-

  b. What is the probability that at most 15 events occur in

  ber of chips in a cookie purchased by the next cus-

  the interval [0, .9907]?

  tomer?

  121. Consider a collection A 1 , … ,A k of mutually exclusive and

  122. Consider a communication source that transmits packets

  exhaustive events, and a random variable X whose distri-

  containing digitized speech. After each transmission, the

  bution depends on which of the A i ’s occurs (e.g., a com-

  receiver sends a message indicating whether the transmis-

  muter might select one of three possible routes from home

  sion was successful or unsuccessful. If a transmission is

  to work, with X representing the commute time). Let

  unsuccessful, the packet is re-sent. Suppose a voice

  E (X uA i ) denote the expected value of X given that the event

  packet can be transmitted a maximum of 10 times.

  A i occurs. Then it can be shown that E(X) 5

  Assuming that the results of successive transmissions are

  oE(XuA i ) ? P(A i ), the weighted average of the individual

  independent of one another and that the probability of any

  “conditional expectations” where the weights are the prob-

  particular transmission being successful is p, determine

  abilities of the partitioning events.

  the probability mass function of the rv X 5 the number of

  a. The expected duration of a voice call to a particular

  times a packet is transmitted. Then obtain an expression

  telephone number is 3 minutes, whereas the expected

  for the expected number of times a packet is transmitted.

  BIBlIOgRaphy

  Johnson, Norman, Samuel Kotz, and Adrienne Kemp, Discrete

  properties of discrete and continuous distributions and

  Univariate Distributions, Wiley, New York, 1992. An

  results for specific distributions.

  encyclopedia of information on discrete distributions.

  Ross, Sheldon, Introduction to Probability Models (10th ed.),

  Olkin, Ingram, Cyrus Derman, and Leon Gleser, Probability

  Academic Press, New York, 2010. A good source of mate-

  Models and Applications (2nd ed.), Macmillan, New York,

  rial on the Poisson process and generalizations, and a nice

  1994. Contains an in-depth discussion of both general

  introduction to other topics in applied probability.

  Continuous Random Variables and Probability

  Distributions