Variance
3.0
2.5
2.0
1.5
1.0
0.5
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 = 0.5, = 0.5
= 5, = 1
= 1, = 3
= 2, = 2
= 2, = 5 a
a a a a
b b b b b
FIGURE 4-28 Beta probability
density functions for selected values of the parameters a and b.
Consider the completion time of a large commercial development. The proportion of the maximum allowed time to complete a task is modeled as a beta random variable with α = 2.5 and β = 1. What is the probability that the proportion of the maximum time exceeds 0.7?
Suppose that X denotes the proportion of the maximum time required to complete the task. The probability is
P X > 0 7 x 1 x 3 5
2 5 1
1 1
0
.
1( ) = Γ α + β ( )
Γ α ( ) Γ β ( ) ( − ) = Γ . ( )
Γ . ( ) Γ (
α− β−
∫ ))
= . ( ) . ( ) . π
( ) . ( ) . π . = − . =
∫
.
.
x x
1 5 0
1
2 5 0 7
1 2 5
2 5 1 5 0 5
1 5 0 5 2 5 1 0 7
.
.
0 0 59 .
Example 4-27
If
a> 1 and β > 1 , the mode (peak of the density) is in the interior of [0, 1] and equals mode = α −
α + β − 1 2
This expression is useful to relate the peak of the density to the parameters. Suppose that the pro- portion of time to complete one task among several follows a beta distribution with α = 2.5 and β = 1. The mode of this distribution is (2.5 1)/(3.5 2) 1 − − = . The mean and variance of a beta dis- tribution can be obtained from the integrals, but the details are left to a Mind-Expanding exercise.
Also, although a beta random variable X is defi ned over the interval [0, 1], a random vari-
able W defi ned over the fi nite interval [ a, b ] can be constructed from W = + − a (b a)X .
4-184. Suppose that X has a beta distribution with parameters α =2.5 and β =2.5. Sketch an approximate graph of the prob- ability density function. Is the density symmetric?
4-185. Suppose that x has a beta distribution with parameters α =2.5 and β =1. Determine the following:
(a) P X
(
<0 25.)
(b) P(
0 25. < < 0.75X)
(c) Mean and variance
4-186. Suppose that X has a beta distribution with parameters α =1 and β =4 2. . Determine the following:
(a) P X
(
<0 25.)
(b) P(
0 5. <X)
(c) Mean and variance 4-187. A European standard value for a low-emission win- dow glazing uses 0.59 as the proportion of solar energy that enters a room. Suppose that the distribution of the proportion of solar energy that enters a room is a beta random variable.(a) Calculate the mode, mean, and variance of the distribution for α =3 and β =1 4. .
(b) Calculate the mode, mean, and variance of the distribution for α =10 and β =6 25. .
(c) Comment on the difference in dispersion in the distribution from parts (a) and (b).
4-188. The length of stay at a hospital emergency department is the sum of the waiting and service times. Let X denote the proportion of time spent waiting and assume a beta distribution with α =10 and β =1. Determine the following:
(a) P X
(
>0 9.)
(b) P X(
<0 5.)
(c) Mean and variance 4-189. The maximum time to complete a task in a project is 2.5 days. Suppose that the completion time as a proportion of this max- imum is a beta random variable with α = 2 and β = 3. What is the probability that the task requires more than two days to complete?FOR SECTION 4-12
Exercises
Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion.
The time to complete a task in a large project is modeled as a generalized beta distribution with minimum and maximum times a = 8 and b = 20 days, respectively, along with mode of m = 16 days. Also, assume that the mean completion time is μ = ( a + 4 m b + / ) 6. Determine the parameters α and β of the generalized beta distribution with these properties.
The values ( a m b , , ) specify the minimum, mode, and maximum times, but the mode value alone does not uniquely deter- mine the two parameters α and β . Consequently, the mean completion time, μ , is assumed to equal μ = ( a + 4 m b + / ) 6.
Here the generalized beta random variable is W = + − a ( b a X ) , where X is a beta random variable. Because the minimum and maximum values for W are 8 and 20, respectively, a = 8 and b = 20. The mean of W is
μ α
= + − = + − α β
a ( b a E X ) ( ) a ( b a ) +
( )
The assumed mean is μ = + ( 8 4 16 ( ) + 20 6 ) / = . 15 333. The mode of W is m = + − a b a −
( ) + − α α β
1 2 with m = 16. These equations can be solved for α and β to obtain
α μ
μ
β α μ
μ
= − − −
− −
= −
−
( )( )
( )( )
( )
a m a b
m b a
b a
2
Therefore,
α β
= . − − −
− . − = .
= . −
( )( ( ) )
( )( )
(
15 333 8 2 16 8 20
16 15 333 20 8 3 665
3 665 20 1 15 333
15 333 8 . 2 333
. − ) = .
Practical Interpretation: The program evaluation and review technique (PERT) widely uses the distribution of W to model the duration of tasks. Therefore, W is said to have a PERT distribution. Notice that we need only specify the minimum, maximum, and mode (most likely time) for a task to specify the distribution. The model assumes that the mean is the function of these three values and allows the α and β parameters to be computed.
Example 4-28
Section 4-12/Beta Distribution 151 4-190. An allele is an alternate form of a gene, and the propor-
tion of alleles in a population is of interest in genetics. An article in BMC Genetics [“Calculating Expected DNA Remnants From Ancient Founding Events in Human Population Genetics” (2008, Vol. 9:66)] used a beta distribution with mean 0.3 and standard deviation 0.17 to model initial allele proportions in a genetic simu- lation. Determine the parameters α and β for this beta distribution.
4-191. Suppose that the construction of a solar power station is initiated. The project’s completion time has not been set due to
uncertainties in financial resources. The completion time for the first phase is modeled with a beta distribution and the minimum, most likely (mode), and maximum completion times for the first phase are 1.0, 1.25, and 2.0 years, respectively. Also, the mean time is assumed to equal μ = +1 4 1 25( . )+ / = .2 6 1 333) . Deter- mine the following in parts (a) and (b):
(a) Parameters α and β of the beta distribution.
(b) Standard deviation of the distribution.
(c) Sketch the probability density function.
Supplemental Exercises
4-192. The probability density function of the time it takes a hematology cell counter to complete a test on a blood sample is f x
( )
= . 0 04for50< x <75 seconds.(a) What percentage of tests requires more than 70 seconds to complete?
(b) What percentage of tests requires less than one minute to complete?
(c) Determine the mean and variance of the time to complete a test on a sample.
4-193. The tensile strength of paper is modeled by a nor- mal distribution with a mean of 35 pounds per square inch and a standard deviation of 2 pounds per square inch.
(a) What is the probability that the strength of a sample is less than 40 lb/in2?
(b) If the specifications require the tensile strength to exceed 30 lb/in2, what proportion of the samples is scrapped?
4-194. The time it takes a cell to divide (called mitosis) is normally distributed with an average time of one hour and a standard deviation of five minutes.
(a) What is the probability that a cell divides in less than 45 minutes?
(b) What is the probability that it takes a cell more than 65 minutes to divide?
(c) By what time have approximately 99% of all cells com- pleted mitosis?
4-195. The length of an injection-molded plastic case that holds magnetic tape is normally distributed with a length of 90.2 millimeters and a standard deviation of 0.1 millimeter.
(a) What is the probability that a part is longer than 90.3 mil- limeters or shorter than 89.7 millimeters?
(b) What should the process mean be set at to obtain the high- est number of parts between 89.7 and 90.3 millimeters?
(c) If parts that are not between 89.7 and 90.3 millimeters are scrapped, what is the yield for the process mean that you selected in part (b)?
Assume that the process is centered so that the mean is 90 milli- meters and the standard deviation is 0.1 millimeter. Suppose that 10 cases are measured, and they are assumed to be independent.
(d) What is the probability that all 10 cases are between 89.7 and 90.3 millimeters?
(e) What is the expected number of the 10 cases that are between 89.7 and 90.3 millimeters?
4-196. The sick-leave time of employees in a firm in a month is normally distributed with a mean of 100 hours and a standard deviation of 20 hours.
(a) What is the probability that the sick-leave time for next month will be between 50 and 80 hours?
(b) How much time should be budgeted for sick leave if the budgeted amount should be exceeded with a probability of only 10%?
4-197. The percentage of people exposed to a bacteria who become ill is 20%. Assume that people are independent. Assume that 1000 people are exposed to the bacteria. Approximate each of the following:
(a) Probability that more than 225 become ill (b) Probability that between 175 and 225 become ill
(c) Value such that the probability that the number of people who become ill exceeds the value is 0.01
4-198. The time to failure (in hours) for a laser in a cytom- etry machine is modeled by an exponential distribution with λ= .0 00004 What is the probability that the time until failure is. (a) At least 20,000 hours? (b) At most 30,000 hours?
(c) Between 20,000 and 30,000 hours?
4-199. When a bus service reduces fares, a particular trip from New York City to Albany, New York, is very popular. A small bus can carry four passengers. The time between calls for tickets is exponentially distributed with a mean of 30 minutes.
Assume that each caller orders one ticket. What is the probabil- ity that the bus is filled in less than three hours from the time of the fare reduction?
4-200. The time between process problems in a manufac- turing line is exponentially distributed with a mean of 30 days.
(a) What is the expected time until the fourth problem?
(b) What is the probability that the time until the fourth prob- lem exceeds 120 days?
4-201. The life of a recirculating pump follows a Weibull dis- tribution with parameters β =2 and δ =700 hours. Determine for parts (a) and (b):
(a) Mean life of a pump (b) Variance of the life of a pump (c) What is the probability that a pump will last longer than its
mean?
Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion.
4-202. The size of silver particles in a photographic emulsion is known to have a log normal distribution with a mean of 0.001 mm and a standard deviation of 0.002 mm.
(a) Determine the parameter values for the lognormal distribution.
(b) What is the probability of a particle size greater than 0.005 mm?
4-203. Suppose that f x
( )
= . −0 5x 1 for 2< x <4. Determine the following:(a) P X <
(
2 5.)
(b) P X >(
3)
(c) P(
2 5. < X <3 5.)
(d) Determine the cumulative distribution function of the ran- dom variable.
(e) Determine the mean and variance of the random variable.
4-204. The time between calls is exponentially distrib- uted with a mean time between calls of 10 minutes.
(a) What is the probability that the time until the first call is less than five minutes?
(b) What is the probability that the time until the first call is between 5 and 15 minutes?
(c) Determine the length of an interval of time such that the probability of at least one call in the interval is 0.90.
(d) If there has not been a call in 10 minutes, what is the proba- bility that the time until the next call is less than 5 minutes?
(e) What is the probability that there are no calls in the inter- vals from 10:00 to 10:05, from 11:30 to 11:35, and from 2:00 to 2:05?
(f) What is the probability that the time until the third call is greater than 30 minutes?
(g) What is the mean time until the fifth call?
4-205. The CPU of a personal computer has a lifetime that is exponentially distributed with a mean lifetime of six years. You have owned this CPU for three years.
(a) What is the probability that the CPU fails in the next three years?
(b) Assume that your corporation has owned 10 CPUs for three years, and assume that the CPUs fail independently. What is the probability that at least one fails within the next three years?
4-206. Suppose that X has a lognormal distribution with parameters θ =0 and ω =2 4. Determine the following:
(a) P
(
10< X <50)
(b) Value for x such that P X < x
( )
= .0 05(c) Mean and variance of X
4-207. Suppose that X has a lognormal distribution and that the mean and variance of X are 50 and 4000, respectively.
Determine the following:
(a) Parameters θ and ω2 of the lognormal distribution (b) Probability that X is less than 150
4-208. Asbestos fibers in a dust sample are identified by an electron microscope after sample preparation. Suppose that the number of fibers is a Poisson random variable and the mean number of fibers per square centimeter of surface dust is 100. A sample of 800 square centimeters of dust is analyzed. Assume that a particular grid cell under the microscope represents 1/160,000 of the sample.
(a) What is the probability that at least one fiber is visible in the grid cell?
(b) What is the mean of the number of grid cells that need to be viewed to observe 10 that contain fibers?
(c) What is the standard deviation of the number of grid cells that need to be viewed to observe 10 that contain fibers?
4-209. Without an automated irrigation system, the height of plants two weeks after germination is normally distributed with a mean of 2.5 centimeters and a standard deviation of 0.5 centimeter.
(a) What is the probability that a plant’s height is greater than 2.25 centimeters?
(b) What is the probability that a plant’s height is between 2.0 and 3.0 centimeters?
(c) What height is exceeded by 90% of the plants?
4-210. With an automated irrigation system, a plant grows to a height of 3.5 centimeters two weeks after germination. Without an automated system, the height is normally distributed with mean and standard deviation 2.5 and 0.5 centimeters, respectively.
(a) What is the probability of obtaining a plant of this height or greater without an automated system?
(b) Do you think the automated irrigation system increases the plant height at two weeks after germination?
4-211. The thickness of a laminated covering for a wood surface is normally distributed with a mean of five millimeters and a standard deviation of 0.2 millimeter.
(a) What is the probability that a covering thickness is more than 5.5 millimeters?
(b) If the specifications require the thickness to be between 4.5 and 5.5 millimeters, what proportion of coverings does not meet specifications?
(c) The covering thickness of 95% of samples is below what value?
4-212. The diameter of the dot produced by a printer is normally distributed with a mean diameter of 0.002 inch.
(a) Suppose that the specifications require the dot diameter to be between 0.0014 and 0.0026 inch. If the probability that a dot meets specifications is to be 0.9973, what standard deviation is needed?
(b) Assume that the standard deviation of the size of a dot is 0.0004 inch. If the probability that a dot meets speci- fications is to be 0.9973, what specifications are needed?
Assume that the specifications are to be chosen symmetri- cally around the mean of 0.002.
4-213. The waiting time for service at a hospital emergency department follows an exponential distribution with a mean of three hours. Determine the following:
(a) Waiting time is greater than four hours
(b) Waiting time is greater than six hours given that you have already waited two hours
(c) Value x (in hours) exceeded with probability 0.25
4-214. The life of a semiconductor laser at a constant power is normally distributed with a mean of 7000 hours and a standard deviation of 600 hours.
(a) What is the probability that a laser fails before 5800 hours?
(b) What is the life in hours that 90% of the lasers exceed?
(c) What should the mean life equal for 99% of the lasers to exceed 10,000 hours before failure?
Section 4-12/Beta Distribution 153
(d) A product contains three lasers, and the product fails if any of the lasers fails. Assume that the lasers fail indepen- dently. What should the mean life equal for 99% of the products to exceed 10,000 hours before failure?
4-215. Continuation of Exercise 4-214. Rework parts (a) and (b). Assume that the lifetime is an exponential random variable with the same mean.
4-216. Continuation of Exercise 4-214. Rework parts (a) and (b). Assume that the lifetime is a lognormal random variable with the same mean and standard deviation.
4-217. A square inch of carpeting contains 50 carpet fib- ers. The probability of a damaged fiber is 0.0001. Assume that the damaged fibers occur independently.
(a) Approximate the probability of one or more damaged fib- ers in one square yard of carpeting.
(b) Approximate the probability of four or more damaged fib- ers in one square yard of carpeting.
4-218. An airline makes 200 reservations for a flight that holds 185 passengers. The probability that a passenger arrives for the flight is 0.9, and the passengers are assumed to be independent.
(a) Approximate the probability that all the passengers who arrive can be seated.
(b) Approximate the probability that the flight has empty seats.
(c) Approximate the number of reservations that the airline should allow so that the probability that everyone who arrives can be seated is 0.95. [Hint: Successively try values for the number of reservations.]
4-219. Suppose that the construction of a solar power station is initiated. The project’s completion time has not been set due to uncertainties in financial resources. The proportion of com- pletion within one year has a beta distribution with parameters α =1 and β =5. Determine the following:
(a) Mean and variance of the proportion completed within one year
(b) Probability that more than half of the project is completed within one year
(c) Proportion of the project that is completed within one year with probability 0.9
4-220. An article in IEEE Journal on Selected Areas in Commu- nications [“Impulse Response Modeling of Indoor Radio Propa- gation Channels” (1993, Vol. 11(7), pp. 967–978)] indicated that the successful design of indoor communication systems requires characterization of radio propagation. The distribution of the amplitude of individual multipath components was well modeled with a lognormal distribution. For one test configuration (with 100 ns delays), the mean amplitude was −24 dB (from the peak) with a standard deviation of 4.1 dB. The amplitude decreased nearly linearly with increased excess delay. Determine the following:
(a) Probability the amplitude exceeds −20 dB (b) Amplitude exceeded with probability 0.05
4-221. Consider the regional right ventricle transverse wall motion in patients with pulmonary hypertension (PH). The right ventricle ejection fraction (EF) is approximately normally distributed with standard deviation of 12 for PH subjects, and
with mean and standard deviation of 56 and 8, respectively, for control subjects.
(a) What is the EF for control subjects exceeded with 99%
probability?
(b) What is the mean for PH subjects such that the probability is 1% that the EF of a PH subject is greater than the value in part (a)?
(c) Comment on how well the control and PH subjects [with the mean determined in part (b)] can be distinguished by EF measurements.
4-222. Provide approximate sketches for beta probability density functions with the following parameters. Comment on any symmetries and show any peaks in the probability density functions in the sketches.
(a) α β= <1 (b) α β= =1. (c) α β= >1.
4-223. Among homeowners in a metropolitan area, 25% recy- cle paper each week. A waste management company services 10,000 homeowners (assumed independent). Approximate the following probabilities:
(a) More than 2600 recycle paper in a week (b) Between 2400 and 2600 recycle paper in a week
(c) Number of customers who recycle paper in a week that is exceeded with probability approximately 0.05
4-224. An article in Journal of Theoretical Biology [“Com- puter Model of Growth Cone Behavior and Neuronal Morpho- genesis” (1995, Vol. 174(4), pp. 381–389)] developed a model for neuronal morphogenesis in which neuronal growth cones have a significant function in the development of the nervous system. This model assumes that the time interval between filopodium formation (a process in growth cone behavior) is exponentially distributed with a mean of 6 time units. Deter- mine the following:
(a) Probability formation requires more than nine time units (b) Probability formation occurs within six to seven time units (c) Formation time exceeded with probability 0.9
4-225. An article in Electric Power Systems Research [“On the Self-Scheduling of a Power Producer in Uncertain Trading Envi- ronments” (2008, Vol. 78(3), pp. 311–317)] considered a self- scheduling approach for a power producer. In addition to price and forced outages, another uncertainty was due to generation reallocations to manage congestions. Generation reallocation was modeled as 110X−60 (with range [− ,60 50 MW/h) where ] X has a beta distribution with parameters α = .3 2 and β = .2 8.
Determine the mean and variance of generation reallocation.
4-226. An article in Electronic Journal of Applied Statistical Analysis [“Survival Analysis of Acute Myocardial Infarction Patients Using Non-Parametric and Parametric Approaches”
(2009, Vol. 2(1), pp. 22–36)] described the use of a Weibull distri- bution to model the survival time of acute myocardial infarction (AMI) patients in a hospital-based retrospective study. The shape and scale parameters for the Weibull distribution in the model were 1.16 and 0.25 years, respectively. Determine the following:
(a) Mean and standard deviation of survival time (b) Probability that a patient survives more than a year (c) Survival time exceeded with probability 0.9
Mind-Expanding Exercises
4-227. The steps in this exercise lead to the probability den- sity function of an Erlang random variable Xwith parameters λ and r, f x
( )
=λrx er−1 −λx/( )
r−1! , x > ,0 and r=1 2, ,….(a) Use the Poisson distribution to express P X
(
>x)
.(b) Use the result from part (a) to determine the cumulative distribution function of X.
(c) Differentiate the cumulative distribution function in part (b) and simplify to obtain the probability density function of X. 4-228. A bearing assembly contains 10 bearings. The bearing diameters are assumed to be independent and normally distrib- uted with a mean of 1.5 millimeters and a standard deviation of 0.025 millimeter. What is the probability that the maximum diameter bearing in the assembly exceeds 1.6 millimeters?
4-229. Let the random variable X denote a measurement from a manufactured product. Suppose that the target value for the measurement is m. For example, X could denote a dimen- sional length, and the target might be 10 millimeters. The quality loss of the process producing the product is defined to be the expected value of k X
(
−m)
2, where k is a constant that relates a deviation from target to a loss measured in dollars.(a) Suppose that X is a continuous random variable with E X
( )
=m and V X( )
= σ2. What is the quality loss of the process?(b) Suppose that X is a continuous random variable with E X
( )
= μ and V X( )
= σ2. What is the quality loss of the process?4-230. The lifetime of an electronic amplifier is modeled as an exponential random variable. If 10% of the amplifiers have a mean of 20,000 hours and the remaining amplifiers have a mean of 50,000 hours, what proportion of the ampli- fiers will fail before 60,000 hours?
4-231. Lack of Memory Property. Show that for an expon- ential random variable X, P X
(
<t1+t2 uX>t1)
=P X(
<t2)
.4-232. Determine the mean and variance of a beta random variable. Use the result that the probability density function integrates to 1. That is,
Γ Γ
Γ( ) ( )
( ) ( )
α β
α β α β
+ =
∫
0 − − −1x 11 x 1 for α> , >0β 0.
4-233. The two-parameter exponential distribution uses a dif- ferent range for the random variable X, namely, 0≤ ≤γ x for a constant γ (and this equals the usual exponential distribution in the special case that γ =0). The probability density func- tion for X is f x( )=λexp[−λ(x−γ)] for 0≤ ≤γ x and 0<λ. Determine the following in terms of the parameters λ and γ: (a) Mean and variance of X. (b) P X( < + /γ 1 λ) 4-234. A process is said to be of six-sigma quality if the pro- cess mean is at least six standard deviations from the nearest specification. Assume a normally distributed measurement.
(a) If a process mean is centered between upper and lower specifications at a distance of six standard deviations from each, what is the probability that a product does not meet specifications? Using the result that 0.000001 equals one part per million, express the answer in parts per million.
(b) Because it is difficult to maintain a process mean centered between the specifications, the probability of a product not meeting specifications is often calculated after assum- ing that the process shifts. If the process mean positioned as in part (a) shifts upward by 1.5 standard deviations, what is the probability that a product does not meet speci- fications? Express the answer in parts per million.
(c) Rework part (a). Assume that the process mean is at a distance of three standard deviations.
(d) Rework part (b). Assume that the process mean is at a distance of three standard deviations and then shifts upward by 1.5 standard deviations.
(e) Compare the results in parts (b) and (d) and comment.
Beta random variable Chi-squared distribution Continuity correction Continuous uniform
distribution
Continuous random variable Continuous uniform random
variable
Cumulative distribution function
Erlang random variable
Exponential random variable Gamma function Gamma random variable Gaussian distribution Lack of memory property-
continuous random variable
Lognormal random variable Mean-continuous random
variable
Mean-function of a continuous random variable
Normal approximation to binomial and Poisson probabilities
Normal random variable Poisson process
Probability density function Probability distribution-
continuous random variable
Standard deviation-continuous random variable
Standardizing
Standard normal random variable
Variance-continuous random variable
Weibull random variable