Page 150 - Applied statistics and probability for engineers

128 Chapter 4/Continuous Random Variables and Probability Distributions

4-87. Assume that a random variable is normally distributed 4-91. A signal in a communication channel is detected when
with a mean of 24 and a standard deviation of 2. Consider an the voltage is higher than 1.5 volts in absolute value. Assume
interval of length one unit that starts at the value a so that the that the voltage is normally distributed with a mean of 0. What
interval is[ , + 1]a a . For what value of a is the probability of is the standard deviation of voltage such that the probability of
the interval greatest? Does the standard deviation affect that a false signal is 0.005?
choice of interval? 4-92. An article in Microelectronics Reliability [“Advanced
4-88. A study by Bechtel et al., 2009, described in the Archives Electronic Prognostics through System Telemetry and Pattern
of Environmental & Occupational Health considered polycy- Recognition Methods” (2007, Vol.47(12), pp. 1865–1873)] pre-
clic aromatic hydrocarbons and immune system function in beef sented an example of electronic prognosis. The objective was to
cattle. Some cattle were near major oil- and gas-producing areas detect faults to decrease the system downtime and the number of
of western Canada. The mean monthly exposure to PM1.0 (par- unplanned repairs in high-reliability systems. Previous measure-
ticulate matter that is < μ1 m in diameter) was approximately ments of the power supply indicated that the signal is normally
μ
3
7.1 g/m with standard deviation 1.5. Assume that the monthly distributed with a mean of 1.5 V and a standard deviation of 0.02 V.
exposure is normally distributed. (a) Suppose that lower and upper limits of the predetermined
(a) What is the probability of a monthly exposure greater than speciications are 1.45 V and 1.55 V, respectively. What is
μ
3
9 g/m ? the probability that a signal is within these speciications?
(b) What is the probability of a monthly exposure between 3 (b) What is the signal value that is exceeded with 95% probability?
and 8 μ g/m ? (c) What is the probability that a signal value exceeds the
3
(c) What is the monthly exposure level that is exceeded with mean by two or more standard deviations?
probability 0.05? 4-93. An article in International Journal of Electrical Power
(d) What value of mean monthly exposure is needed so that & Energy Systems [“Stochastic Optimal Load Flow Using a
the probability of a monthly exposure more than 9 μ g/m is Combined Quasi–Newton and Conjugate Gradient Technique”
3
0.01? (1989, Vol.11(2), pp. 85–93)] considered the problem of opti-
4-89. An article in Atmospheric Chemistry and Physics “Rela- mal power low in electric power systems and included the
tionship Between Particulate Matter and Childhood Asthma— effects of uncertain variables in the problem formulation. The
Basis of a Future Warning System for Central Phoenix” (2012, method treats the system power demand as a normal random
Vol. 12, pp. 2479–2490)] reported the use of PM10 (particulate variable with 0 mean and unit variance.
matter <10 μm diameter) air quality data measured hourly from (a) What is the power demand value exceeded with 95%
sensors in Phoenix, Arizona. The 24-hour (daily) mean PM10 probability?
μ
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for a centrally located sensor was 50.9 g/m with a standard (b) What is the probability that the power demand is positive?
deviation of 25.0. Assume that the daily mean of PM10 is nor- (c) What is the probability that the power demand is more than
mally distributed. – 1 and less than 1?
(a) What is the probability of a daily mean of PM10 greater 4-94. An article in the Journal of Cardiovascular Magnetic
μ
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than 100 g/m ? Resonance [“Right Ventricular Ejection Fraction Is Better
(b) What is the probability of a daily mean of PM10 less than Relected by Transverse Rather Than Longitudinal Wall Motion
μ
3
25 g/m ? in Pulmonary Hypertension” (2010, Vol.12(35)] discussed a
(c) What daily mean of PM10 value is exceeded with prob- study of the regional right ventricle transverse wall motion in
ability 5%? patients with pulmonary hypertension (PH). The right ventricle
4-90. The length of stay at a speciic emergency department ejection fraction (EF) was approximately normally distributed
in Phoenix, Arizona, in 2009 had a mean of 4.6 hours with with a mean and a standard deviation of 36 and 12, respec-
a standard deviation of 2.9. Assume that the length of stay is tively, for PH subjects, and with mean and standard deviation
normally distributed. of 56 and 8, respectively, for control subjects.
(a) What is the probability of a length of stay greater than 10 hours? (a) What is the EF for PH subjects exceeded with 5% probability?
(b) What length of stay is exceeded by 25% of the visits? (b) What is the probability that the EF of a control subject is
(c) From the normally distributed model, what is the probabil- less than the value in part (a)?
ity of a length of stay less than 0 hours? Comment on the (c) Comment on how well the control and PH subjects can be
normally distributed assumption in this example. distinguished by EF measurements.

4-7 Normal Approximation to the Binomial and

Poisson Distributions

We began our section on the normal distribution with the central limit theorem and the
normal distribution as an approximation to a random variable with a large number of trials.
Consequently, it should not be surprising to learn that the normal distribution can be used