Page 175 - Applied statistics and probability for engineers

Section 4-12/Beta Distribution 153

(d) A product contains three lasers, and the product fails if with mean and standard deviation of 56 and 8, respectively, for
any of the lasers fails. Assume that the lasers fail indepen- control subjects.
dently. What should the mean life equal for 99% of the (a) What is the EF for control subjects exceeded with 99%
products to exceed 10,000 hours before failure? probability?
4-215. Continuation of Exercise 4-214. Rework parts (a) and (b) What is the mean for PH subjects such that the probability
(b). Assume that the lifetime is an exponential random variable is 1% that the EF of a PH subject is greater than the value
with the same mean. in part (a)?
4-216. Continuation of Exercise 4-214. Rework parts (a) and (c) Comment on how well the control and PH subjects [with
(b). Assume that the lifetime is a lognormal random variable the mean determined in part (b)] can be distinguished by
with the same mean and standard deviation. EF measurements.
4-217. A square inch of carpeting contains 50 carpet ib-
ers. The probability of a damaged iber is 0.0001. Assume that 4-222. Provide approximate sketches for beta probability
the damaged ibers occur independently. density functions with the following parameters. Comment on
(a) Approximate the probability of one or more damaged ib- any symmetries and show any peaks in the probability density
ers in one square yard of carpeting. functions in the sketches. β β
β
(b) Approximate the probability of four or more damaged ib- (a) α = <1 (b) α = = 1. (c) α = >1.
ers in one square yard of carpeting. 4-223. Among homeowners in a metropolitan area, 25% recy-
cle paper each week. A waste management company services
4-218. An airline makes 200 reservations for a light that 10,000 homeowners (assumed independent). Approximate the
holds 185 passengers. The probability that a passenger arrives following probabilities:
for the light is 0.9, and the passengers are assumed to be (a) More than 2600 recycle paper in a week
independent. (b) Between 2400 and 2600 recycle paper in a week
(a) Approximate the probability that all the passengers who (c) Number of customers who recycle paper in a week that is
arrive can be seated. exceeded with probability approximately 0.05
(b) Approximate the probability that the light has empty seats.
(c) Approximate the number of reservations that the airline 4-224. An article in Journal of Theoretical Biology [“Com-
should allow so that the probability that everyone who puter Model of Growth Cone Behavior and Neuronal Morpho-
arrives can be seated is 0.95. [Hint: Successively try values genesis” (1995, Vol. 174(4), pp. 381–389)] developed a model
for the number of reservations.] for neuronal morphogenesis in which neuronal growth cones
have a signiicant function in the development of the nervous
4-219. Suppose that the construction of a solar power station system. This model assumes that the time interval between
is initiated. The project’s completion time has not been set due ilopodium formation (a process in growth cone behavior) is
to uncertainties in inancial resources. The proportion of com- exponentially distributed with a mean of 6 time units. Deter-
pletion within one year has a beta distribution with parameters mine the following:
α = 1 and β = 5. Determine the following: (a) Probability formation requires more than nine time units
(a) Mean and variance of the proportion completed within (b) Probability formation occurs within six to seven time units
one year (c) Formation time exceeded with probability 0.9
(b) Probability that more than half of the project is completed
within one year 4-225. An article in Electric Power Systems Research [“On the
(c) Proportion of the project that is completed within one year Self-Scheduling of a Power Producer in Uncertain Trading Envi-
with probability 0.9 ronments” (2008, Vol. 78(3), pp. 311–317)] considered a self-
scheduling approach for a power producer. In addition to price
4-220. An article in IEEE Journal on Selected Areas in Commu- and forced outages, another uncertainty was due to generation
nications [“Impulse Response Modeling of Indoor Radio Propa- reallocations to manage congestions. Generation reallocation
gation Channels” (1993, Vol. 11(7), pp. 967–978)] indicated that was modeled as 110X − 60 (with range [− ,60 50 ] MW/h) where
the successful design of indoor communication systems requires X has a beta distribution with parameters α = .2 and β = .8.
3
2
characterization of radio propagation. The distribution of the Determine the mean and variance of generation reallocation.
amplitude of individual multipath components was well modeled 4-226. An article in Electronic Journal of Applied Statistical
with a lognormal distribution. For one test coniguration (with 100 Analysis [“Survival Analysis of Acute Myocardial Infarction
ns delays), the mean amplitude was −24 dB (from the peak) with Patients Using Non-Parametric and Parametric Approaches”
a standard deviation of 4.1 dB. The amplitude decreased nearly (2009, Vol. 2(1), pp. 22–36)] described the use of a Weibull distri-
linearly with increased excess delay. Determine the following: bution to model the survival time of acute myocardial infarction
(a) Probability the amplitude exceeds −20 dB (AMI) patients in a hospital-based retrospective study. The shape
(b) Amplitude exceeded with probability 0.05 and scale parameters for the Weibull distribution in the model
4-221. Consider the regional right ventricle transverse wall were 1.16 and 0.25 years, respectively. Determine the following:
motion in patients with pulmonary hypertension (PH). The (a) Mean and standard deviation of survival time
right ventricle ejection fraction (EF) is approximately normally (b) Probability that a patient survives more than a year
distributed with standard deviation of 12 for PH subjects, and (c) Survival time exceeded with probability 0.9