Page 167 - Applied statistics and probability for engineers

Section 4-11/Lognormal Distribution 145

4-158. Assume that the life of a packaged magnetic disk (a) P X( > 3500)
(
exposed to corrosive gases has a Weibull distribution with (b) P X > 3500) for an exponential random variable with the
0
β = .5 and the mean life is 600 hours. Determine the following: same mean as the Weibull distribution
(a) Probability that a disk lasts at least 500 hours. (c) Comment on the probability that the lifetime exceeds 3500
(b) Probability that a disk fails before 400 hours. hours under the Weibull and exponential distributions.
4-159. The life (in hours) of a magnetic resonance imag- 4-166. An article in Electronic Journal of Applied Statistical
ing machine (MRI) is modeled by a Weibull distribution with Analysis [“Survival Analysis of Dialysis Patients Under Para-
parameters β = 2 and δ = 500 hours. Determine the following: metric and Non-Parametric Approaches” (2012, Vol. 5(2), pp.
(a) Mean life of the MRI 271–288)] modeled the survival time of dialysis patients with
(b) Variance of the life of the MRI chronic kidney disease with a Weibull distribution. The mean
(c) Probability that the MRI fails before 250 hours. and standard deviation of survival time were 16.01 and 11.66
4-160. An article in the Journal of the Indian Geophysi- months, respectively. Determine the following:
cal Union titled “Weibull and Gamma Distributions for Wave (a) Shape and scale parameters of this Weibull distribution
Parameter Predictions” (2005, Vol. 9, pp. 55–64) described the (b) Probability that survival time is more than 48 months
use of the Weibull distribution to model ocean wave heights. (c) Survival time exceeded with 90% probability
Assume that the mean wave height at the observation station is 4-167. An article in Proceeding of the 33rd International
2.5 m and the shape parameter equals 2. Determine the stand- ACM SIGIR Conference on Research and Development in
ard deviation of wave height. Information Retrieval [“Understanding Web Browsing Behav-
4-161. An article in the Journal of Geophysical Research iors Through Weibull Analysis of Dwell Time” (2010, p. 379l–
[“Spatial and Temporal Distributions of U.S. of Winds and Wind 386)] proposed that a Weibull distribution can be used to model
Power at 80 m Derived from Measurements” (2003, vol. 108)] Web page dwell time (the length of time a Web visitor spends
considered wind speed at stations throughout the United States. on a Web page). For a speciic Web page, the shape and scale
A Weibull distribution can be used to model the distribution of parameters are 1 and 300 seconds, respectively. Determine the
wind speeds at a given location. Every location is characterized following:
by a particular shape and scale parameter. For a station at Ama- (a) Mean and variance of dwell time
rillo, Texas, the mean wind speed at 80 m (the hub height of large (b) Probability that a Web user spends more than four minutes
wind turbines) in 2000 was 10.3 m/s with a standard deviation of on this Web page
4.9 m/s. Determine the shape and scale parameters of a Weibull
distribution with these properties. (c) Dwell time exceeded with probability 0.25
4-162. Suppose that X has a Weibull distribution with β = 2 4-168. An article in Financial Markets Institutions and Instru-
and δ = 8 6. . Determine the following: ments [“Pricing Reinsurance Contracts on FDIC Losses” (2008,
(a) P X( < 10) (b) P X > 9) (c) P(8 < X 11) Vol. 17(3)] modeled average annual losses (in billions of dollars)
<
(
(d) Value for x such that P X( > x) = 0.9 of the Federal Deposit Insurance Corporation (FDIC) with a
4-163. Suppose that the lifetime of a component (in hours) is Weibull distribution with parameters δ = .1 9317 and β = .0 8472.
modeled with a Weibull distribution with β = 2 and δ = 4000. Determine the following:
Determine the following in parts (a) and (b): (a) Probability of a loss greater than $2 billion
(
(a) P X( > 3000) (b) P X > 6000u X > 3000) (b) Probability of a loss between $2 and $4 billion
(c) Comment on the probabilities in the previous parts com- (c) Value exceeded with probability 0.05
pared to the results for an exponential distribution. (d) Mean and standard deviation of loss
4-164. Suppose that the lifetime of a component (in hours), X is 4-169. An article in IEEE Transactions on Dielectrics and
modeled with a Weibull distribution with β = 0 5. and δ = 4000. Electrical Insulation [“Statistical Analysis of the AC Break-
Determine the following in parts (a) and (b): down Voltages of Ester Based Transformer Oils” (2008, Vol.
(
(a) P X( > 3500) (b) P X > 6000 X > 3000) 15(4))] used Weibull distributions to model the breakdown
(c) Comment on the probabilities in the previous parts com- voltage of insulators. The breakdown voltage is the mini-
pared to the results for an exponential distribution. mum voltage at which the insulator conducts. For 1 mm of
(d) Comment on the role of the parameter β in a lifetime model natural ester, the 1% probability of breakdown voltage is
with the Weibull distribution. approximately 26 kV, and the 7% probability is approximately
4-165. Suppose that X has a Weibull distribution with β = 2 31.6 kV. Determine the parameters δ and β of the Weibull
and δ = 2000. Determine the following in parts (a) and (b): distribution.

4-11 Lognormal Distribution

Variables in a system sometimes follow an exponential relationship as x = exp w ( ). If the expo-
nent is a random variable W,then X = exp W ( ) is a random variable with a distribution of
interest. An important special case occurs when W has a normal distribution. In that case, the