28 Project managers often use a method labeled PERT—for program evaluation and

Example 4.28 Project managers often use a method labeled PERT—for program evaluation and

  review technique—to coordinate the various activities making up a large project. (One successful application was in the construction of the Apollo spacecraft.) A stan- dard assumption in PERT analysis is that the time necessary to complete any partic-

  ular activity once it has been started has a beta distribution with A5 the optimistic time (if everything goes well) and B5 the pessimistic time (if everything goes

  badly). Suppose that in constructing a single-family house, the time X (in days) nec- essary for laying the foundation has a beta distribution with

  A 5 2, B 5 5, a 5 2 ,

  and b53 . Then a(a 1 b) 5 .4 , so E(X) 5 2 1 (3)(.4) 5 3.2 . For these values

  of and , the pdf of X is a simple polynomial function. The probability that it takes a b at most 3 days to lay the foundation is

  4! x22 52x 2

  5 (x 2 2)(5 2 x) 2 dx 5

  5 5 .407 ■

  27 3 2 27 4 27 The standard beta distribution is commonly used to model variation in the pro-

  portion or percentage of a quantity occurring in different samples, such as the pro- portion of a 24-hour day that an individual is asleep or the proportion of a certain element in a chemical compound.

  EXERCISES Section 4.5 (72–86)

  72. The lifetime X (in hundreds of hours) of a certain type of

  X 2 3.5 over the minimum has a Weibull distribution with

  vacuum tube has a Weibull distribution with parameters

  parameters and (see a52 b 5 1.5 “Practical Applications

  a52 and b53 . Compute the following:

  of the Weibull Distribution,” Industrial Quality Control,

  a. E(X) and V(X)

  Aug. 1964: 71–78).

  b. P(X 6)

  a. What is the cdf of X?

  c. P(1.5 X 6)

  b. What are the expected return time and variance of return

  (This Weibull distribution is suggested as a model for time

  time? [Hint: First obtain E(X 2 3.5) and V(X 2 3.5) .]

  in service in “On the Assessment of Equipment Reliability:

  c. Compute . P(X . 5)

  Trading Data Collection Costs for Precision,” J. of Engr.

  d. Compute . P(5 X 8)

  Manuf., 1991: 105–109.)

  75. Let X have a Weibull distribution with the pdf from

  Expression (4.11). Verify that m5b (1 1 1a) . [Hint: In

  73. The authors of the article “A Probabilistic Insulation Life

  Model for Combined Thermal-Electrical Stresses” (IEEE

  the integral for E(X), make the change of variable

  a Trans. on Elect. Insulation, 1985: 519–522) state that “the 1a y 5 (xb) , so that x 5 by .] Weibull distribution is widely used in statistical problems

  76. a. In Exercise 72, what is the median lifetime of such

  relating to aging of solid insulating materials subjected to

  tubes? [Hint: Use Expression (4.12).]

  aging and stress.” They propose the use of the distribution

  b. In Exercise 74, what is the median return time?

  as a model for time (in hours) to failure of solid insulating

  c. If X has a Weibull distribution with the cdf from

  specimens subjected to AC voltage. The values of the

  Expression (4.12), obtain a general expression for the

  parameters depend on the voltage and temperature; sup-

  (100p)th percentile of the distribution.

  pose a 5 2.5 and b 5 200 (values suggested by data in the

  d. In Exercise 74, the company wants to refuse to accept

  article).

  returns after t weeks. For what value of t will only 10

  a. What is the probability that a specimen’s lifetime is at

  of all returns be refused?

  most 250? Less than 250? More than 300?

  77. The authors of the paper from which the data in Exercise

  b. What is the probability that a specimen’s lifetime is

  1.27 was extracted suggested that a reasonable probability

  between 100 and 250?

  model for drill lifetime was a lognormal distribution with

  c. What value is such that exactly 50 of all specimens

  m 5 4.5 and . s 5 .8

  have lifetimes exceeding that value?

  a. What are the mean value and standard deviation of lifetime?

  74. Let X5 the time (in 10 21 weeks) from shipment of a defec-

  b. What is the probability that lifetime is at most 100?

  tive product until the customer returns the product. Suppose

  c. What is the probability that lifetime is at least 200?

  that the minimum return time is g 5 3.5 and that the excess

  Greater than 200?

  CHAPTER 4 Continuous Random Variables and Probability Distributions

  78. The article “On Assessing the Accuracy of Offshore Wind

  b. Compute . P(X . 125)

  Turbine Reliability-Based Design Loads from the

  c. Compute . P(110 X 125)

  Environmental Contour Method” (Intl. J. of Offshore and

  d. What is the value of median ductile strength?

  Polar Engr., 2005: 132–140) proposes the Weibull distribu-

  e. If ten different samples of an alloy steel of this type were

  tion with a 5 1.817 and

  b 5 .863 as a model for 1-hour

  subjected to a strength test, how many would you expect

  significant wave height (m) at a certain site.

  to have strength of at least 125?

  a. What is the probability that wave height is at most .5 m?

  f. If the smallest 5 of strength values were unacceptable,

  b. What is the probability that wave height exceeds its

  what would the minimum acceptable strength be?

  mean value by more than one standard deviation?

  82. The article “The Statistics of Phytotoxic Air Pollutants”

  c. What is the median of the wave-height distribution?

  (J. of Royal Stat. Soc., 1989: 183–198) suggests the lognor-

  d. For 0,p,1 , give a general expression for the 100pth

  mal distribution as a model for SO 2 concentration above a

  percentile of the wave-height distribution.

  certain forest. Suppose the parameter values are m 5 1.9

  79. Nonpoint source loads are chemical masses that travel to the

  and . s 5 .9

  main stem of a river and its tributaries in flows that are dis-

  a. What are the mean value and standard deviation of con-

  tributed over relatively long stream reaches, in contrast to

  centration?

  those that enter at well-defined and regulated points. The

  b. What is the probability that concentration is at most 10?

  article “Assessing Uncertainty in Mass Balance Calculation

  Between 5 and 10?

  of River Nonpoint Source Loads” (J. of Envir. Engr., 2008:

  83. What condition on a and b is necessary for the standard

  247–258) suggested that for a certain time period and loca-

  beta pdf to be symmetric?

  tion, X5 nonpoint source load of total dissolved solids could be modeled with a lognormal distribution having

  84. Suppose the proportion X of surface area in a randomly

  mean value 10,281 kgdaykm and a coefficient of variation

  selected quadrat that is covered by a certain plant has a stan-

  CV 5 .40 (CV 5 s X m X ) .

  dard beta distribution with a55 and b52 .

  a. What are the mean value and standard deviation of

  a. Compute E(X) and V(X).

  ln(X)?

  b. Compute . P(X .2)

  b. What is the probability that X is at most 15,000

  c. Compute . P(.2 X .4)

  kgdaykm?

  d. What is the expected proportion of the sampling region

  c. What is the probability that X exceeds its mean value,

  not covered by the plant?

  and why is this probability not .5?

  85. Let X have a standard beta density with parameters and . a b

  d. Is 17,000 the 95th percentile of the distribution?

  a. Verify the formula for E(X) given in the section.

  | m

  b. Compute . E[(1 2 X) 80. a. Use Equation (4.13) to write a formula for the median m ] If X represents the proportion of

  of the lognormal distribution. What is the median for the

  a substance consisting of a particular ingredient, what is

  load distribution of Exercise 79?

  the expected proportion that does not consist of this

  b. Recalling that z a is our notation for the

  per-

  ingredient?

  centile of the standard normal distribution, write an

  86. Stress is applied to a 20-in. steel bar that is clamped in a

  expression for the 100(1 2 a) percentile of the lognor-

  fixed position at each end. Let Y5 the distance from the

  mal distribution. In Exercise 79, what value will load

  left end at which the bar snaps. Suppose Y20 has a standard

  exceed only 1 of the time?

  beta distribution with E(Y) 5 10 and V(Y) 5 100 7 .

  81. A theoretical justification based on a certain material fail-

  a. What are the parameters of the relevant standard beta

  ure mechanism underlies the assumption that ductile

  distribution?

  strength X of a material has a lognormal distribution.

  b. Compute . P(8 Y 12)

  Suppose the parameters are m55 and s 5 .1 .

  c. Compute the probability that the bar snaps more than

  a. Compute E(X) and V(X).

  2 in. from where you expect it to.