EXERCISES Section 3.6 (79-93)

79

The article “Expectation Analysis of the Probability of Failure for Water Supply Pipes” (J. of Pipeline Systems Engr. and Practice, May 2012: 36-46) proposed using the Poisson distribution to model the number of failures in pipelines of various types. Suppose that for cast-iron pipe of a particular length, the expected number of failures is 1 (very close to one of the cases considered in the article). Then $X$ , the number of failures, has a Poisson distribution with $\mu =1$ .

a. Obtain $P\left(X\le 5\right)$ by using Appendix Table A.2.

b. Determine $P\left(X=2\right)$ first from the pmf formula and then from Appendix Table A.2.

c. Determine $P\left(2\le X\le 4\right)$ .

d. What is the probability that $X$ exceeds its mean value by more than one standard deviation?

80

Let $X$ be the number of material anomalies occurring in a particular region of an aircraft gas-turbine disk. The article “Methodology for Probabilistic Life Prediction of Multiple-Anomaly Materials” (Amer. Inst. of Aeronautics and Astronautics J., 2006: 787-793) proposes a Poisson distribution for $X$ . Suppose that $\mu =4$ .

a. Compute both $P\left(X\le 4\right)$ and $P\left(X<4\right)$ .

b. Compute $P\left(4\le X\le 8\right)$ .

c. Compute $P\left(8\le X\right)$ .

d. What is the probability that the number of anomalies exceeds its mean value by no more than one standard deviation?

81

Suppose that the number of drivers who travel between a particular origin and destination during a designated time period has a Poisson distribution with parameter $\mu =20$ (suggested in the article “Dynamic Ride Sharing: Theory and Practice,” J. of Transp. Engr., 1997: 308-312). What is the probability that the number of drivers will

a. Be at most 10 ?

b. Exceed 20?

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

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

82

Consider writing onto a computer disk and then sending it through a certifier that counts the number of missing pulses. Suppose this number $X$ has a Poisson distribution with parameter $\mu =.2$ . (Suggested in “Average Sample Number for Semi-Curtailed Sampling Using the Poisson Distribution,” J. Quality Technology, 1983: 126-129.)

a. What is the probability that a disk has exactly one missing pulse?

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

c. If two disks are independently selected, what is the probability that neither contains a missing pulse?

83

An article in the Los Angeles Times (Dec. 3, 1993) reports that 1 in 200 people carry the defective gene that causes inherited colon cancer. In a sample of 1000 individuals, what is the approximate distribution of the number who carry this gene? Use this distribution to calculate the approximate probability that

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

b. At least 8 carry the gene.

84

The Centers for Disease Control and Prevention reported in 2012 that 1 in 88 American children had been diagnosed with an autism spectrum disorder (ASD).

a. If a random sample of 200 American children is selected, what are the expected value and standard deviation of the number who have been diagnosed with ASD?

b. Referring back to (a), calculate the approximate probability that at least 2 children in the sample have been diagnosed with ASD?

c. If the sample size is 352 , what is the approximate probability that fewer than 5 of the selected children have been diagnosed with ASD?

85

Suppose small aircraft arrive at a certain airport according to a Poisson process with rate $\alpha =8$ per hour, so that the number of arrivals during a time period of $t$ hours is a Poisson rv with parameter $\mu =8t$ .

a. What is the probability that exactly 6 small aircraft arrive during a 1-hour period? At least 6? At least 10 ?

b. What are the expected value and standard deviation of the number of small aircraft that arrive during a 90-min period?

c. What is the probability that at least 20 small aircraft arrive during a 2.5-hour period? That at most 10 arrive during this period?

86

Organisms are present in ballast water discharged from a ship according to a Poisson process with a concentration of 10 organisms $/{\mathrm{m}}^3$ [the article “Counting at Low Concentrations: The Statistical Challenges of Verifying Ballast Water Discharge Standards” (Ecological Applications, 2013: 339-351) considers using the Poisson process for this purpose].

a. What is the probability that one cubic meter of discharge contains at least 8 organisms?

b. What is the probability that the number of organisms in $1.5{\mathrm{m}}^3$ of discharge exceeds its mean value by more than one standard deviation?

c. For what amount of discharge would the probability of containing at least 1 organism be .999 ?

87

The number of requests for assistance received by a towing service is a Poisson process with rate $\alpha =4$ per hour.

a. Compute the probability that exactly ten requests are received during a particular 2-hour period.

b. If the operators of the towing service take a 30-min break for lunch, what is the probability that they do not miss any calls for assistance?

c. How many calls would you expect during their break?

88

In proof testing of circuit boards, the probability that any particular diode will fail is .01 . Suppose a circuit board contains 200 diodes.

a. How many diodes would you expect to fail, and what is the standard deviation of the number that are expected to fail?

b. What is the (approximate) probability that at least four diodes will fail on a randomly selected board?

c. If five boards are shipped to a particular customer, how likely is it that at least four of them will work properly? (A board works properly only if all its diodes work.)

89

The article “Reliability-Based Service-Life Assessment of Aging Concrete Structures” (J. Structural Engr., 1993: 1600-1621) suggests that a Poisson process can be used to represent the occurrence of structural loads over time. Suppose the mean time between occurrences of loads is .5 year.

a. How many loads can be expected to occur during a 2-year period?

b. What is the probability that more than five loads occur during a 2-year period?

c. How long must a time period be so that the probability of no loads occurring during that period is at most .1 ?

90

Let $X$ have a Poisson distribution with parameter $\mu$ . Show that $E\left(X\right)=\mu$ directly from the definition of expected value. Hint: The first term in the sum equals 0 , and then $x$ can be canceled. Now factor out $\mu$ and show that what is left sums to 1.

91

Suppose that trees are distributed in a forest according to a two-dimensional Poisson process with parameter $\alpha$ , the expected number of trees per acre, equal to 80 .

a. What is the probability that in a certain quarter-acre plot, there will be at most 16 trees?

b. If the forest covers 85,000 acres, what is the expected number of trees in the forest?

c. Suppose you select a point in the forest and construct a circle of radius . 1 mile. Let $X=$ the number of trees within that circular region. What is the pmf of $X$ ? Hint: 1 sq mile $=640$ acres.

92

Automobiles arrive at a vehicle equipment inspection station according to a Poisson process with rate $\alpha =10$ per hour. Suppose that with probability . 5 an arriving vehicle will have no equipment violations.

a. What is the probability that exactly ten arrive during the hour and all ten have no violations?

b. For any fixed $y\ge 10$ , what is the probability that $y$ arrive during the hour, of which ten have no violations?

c. What is the probability that ten “no-violation” cars arrive during the next hour? Hint: Sum the probabilities in part (b) from $y=10$ to $\infty$ .

93

a. In a Poisson process, what has to happen in both the time interval $\left(0,t\right)$ and the interval $\left(t,t+\Delta t\right)$ so that no events occur in the entire interval $\left(0,t+\Delta t\right)$ ? Use this and Assumptions 1-3 to write a relationship between $P_0\left(t+\Delta t\right)$ and $P_0\left(t\right)$ .

b. Use the result of part (a) to write an expression for the difference $P_0\left(t+\Delta t\right)-P_0\left(t\right)$ . Then divide by $\Delta t$ and let $\Delta t\to 0$ to obtain an equation involving $\left(d/dt\right)P_0\left(t\right)$ , the derivative of $P_0\left(t\right)$ with respect to $t$ .

c. Verify that $P_0\left(t\right)=e^{-\alpha t}$ satisfies the equation of part (b).

d. It can be shown in a manner similar to parts (a) and (b) that the $P_k\left(t\right)\mathrm{s}$ must satisfy the system of differential equations $\frac{d}{dt}{P}_k\left(t\right)=\alpha P_{k-1}\left(t\right)-\alpha P_k\left(t\right)$ for $k=1,2,3,\dots$

Verify that $P_k\left(t\right)=e^{-\alpha t}{\left(\alpha t\right)}^k/k!$ satisfies the system. (This is actually the only solution.)