EXERCISES Section 4.5 (72-86)

72

The lifetime $X$ (in hundreds of hours) of a certain type of vacuum tube has a Weibull distribution with parameters $\alpha =2$ and $\beta =3$ . Compute the following:

a. $E\left(X\right)$ and $V\left(X\right)$

b. $P\left(X\le 6\right)$

c. $P\left(1.5\le X\le 6\right)$

(This Weibull distribution is suggested as a model for time in service in “On the Assessment of Equipment Reliability: Trading Data Collection Costs for Precision,” J. of Engr: Manuf., 1991: 105-109.)

73

The authors of the article “A Probabilistic Insulation Life Model for Combined Thermal-Electrical Stresses” (IEEE Trans. on Elect. Insulation, 1985: 519-522) state that “the Weibull distribution is widely used in statistical problems relating to aging of solid insulating materials subjected to aging and stress.” They propose the use of the distribution as a model for time (in hours) to failure of solid insulating specimens subjected to AC voltage. The values of the parameters depend on the voltage and temperature; suppose $\alpha =2.5$ and $\beta =200$ (values suggested by data in the article).

a. What is the probability that a specimen’s lifetime is at most 250 ? Less than 250 ? More than 300 ?

b. What is the probability that a specimen’s lifetime is between 100 and 250 ?

c. What value is such that exactly $50\%$ of all specimens have lifetimes exceeding that value?

74

Once an individual has been infected with a certain disease, let $X$ represent the time (days) that elapses before the individual becomes infectious. The article “The Probability of Containment for Multitype Branching Process Models for Emerging Epidemics” (J. of Applied Probability, 2011: 173-188) proposes a Weibull distribution with $\alpha =2.2,\beta =1.1$ , and $\gamma =.5$ (refer to Example 4.26).

a. Calculate $P\left(1<X<2\right)$ .

b. Calculate $P\left(X>1.5\right)$ .

c. What is the 90th percentile of the distribution?

d. What are the mean and standard deviation of $X$ ?

75

Let $X$ have a Weibull distribution with the pdf from Expression (4.11). Verify that $\mu =\beta \Gamma \left(1+1/\alpha \right)$ .

• Hint: In the integral for $E\left(X\right)$ , make the change of variable $y={\left(x/\beta \right)}^{\alpha }$, so that $x=\beta y^{1/\alpha }$

76

The article “The Statistics of Phytotoxic Air Pollutants” (J. of Royal Stat. Soc., 1989: 183-198) suggests the lognormal distribution as a model for ${\mathrm{SO}}_2$ concentration above a certain forest. Suppose the parameter values are $\mu =1.9$ and $\sigma =.9$ .

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

b. What is the probability that concentration is at most 10? Between 5 and 10 ?

77

The authors of the article from which the data in Exercise 1.27 was extracted suggested that a reasonable probability model for drill lifetime was a lognormal distribution with $\mu =4.5$ and $\sigma =.8$ .

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

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

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

78

The article “On Assessing the Accuracy of Offshore Wind Turbine Reliability-Based Design Loads from the Environmental Contour Method” (Intl. J. of Offshore and Polar Engr., 2005: 132-140) proposes the Weibull distribution with $\alpha =1.817$ and $\beta =.863$ as a model for 1-hour significant wave height $\left(\mathrm{m}\right)$ at a certain site.

a. What is the probability that wave height is at most $.5\mathrm{m}$ ?

b. What is the probability that wave height exceeds its mean value by more than one standard deviation?

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

d. For $0<p<1$ , give a general expression for the $100p$ th percentile of the wave-height distribution.

79

Nonpoint source loads are chemical masses that travel to the main stem of a river and its tributaries in flows that are distributed over relatively long stream reaches, in contrast to those that enter at well-defined and regulated points. The article “Assessing Uncertainty in Mass Balance Calculation of River Nonpoint Source Loads” (J. of Envir. Engr., 2008: 247-258) suggested that for a certain time period and location, $X=$ nonpoint source load of total dissolved solids could be modeled with a lognormal distribution having mean value 10,281 $\mathrm{kg}/$ day $/\mathrm{km}$ and a coefficient of variation $CV=.40(CV=$ ${\sigma }_X/{\mu }_X)$ .

a. What are the mean value and standard deviation of $\ln \left(X\right)$ ?

b. What is the probability that $X$ is at most 15,000 $\mathrm{kg}/$ day $/\mathrm{km}$ ?

c. What is the probability that $X$ exceeds its mean value, and why is this probability not .5 ?

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

80

a. Use Equation (4.13) to write a formula for the median $\tilde{\mu }$ of the lognormal distribution. What is the median for the load distribution of Exercise 79?

b. Recalling that $z_{\alpha }$ is our notation for the $100\left(1-\alpha \right)$ percentile of the standard normal distribution, write an expression for the $100\left(1-\alpha \right)$ percentile of the lognormal distribution. In Exercise 79, what value will load exceed only $1\%$ of the time?

81

Sales delay is the elapsed time between the manufacture of a product and its sale. According to the article “Warranty Claims Data Analysis Considering Sales Delay” (Quality and Reliability Engr. Intl., 2013: 113-123), it is quite common for investigators to model sales delay using a lognormal distribution. For a particular product, the cited article proposes this distribution with parameter values $\mu =2.05$ and ${\sigma }^2=.06$ (here the unit for delay is months).

a. What are the variance and standard deviation of delay time?

b. What is the probability that delay time exceeds 12 months?

c. What is the probability that delay time is within one standard deviation of its mean value?

d. What is the median of the delay time distribution?

e. What is the 99th percentile of the delay time distribution?

f. Among 10 randomly selected such items, how many would you expect to have a delay time exceeding 8 months?

82

As in the case of the Weibull and Gamma distributions, the lognormal distribution can be modified by the introduction of a third parameter $\gamma$ such that the pdf is shifted to be positive only for $x>\gamma$ . The article cited in Exercise 4.39 suggested that a shifted lognormal distribution with shift (i.e., threshold) $=1.0$ , mean value $=$ 2.16, and standard deviation $=1.03$ would be an appropriate model for the rv $X=$ maximum-to-average depth ratio of a corrosion defect in pressurized steel.

a. What are the values of $\mu$ and $\sigma$ for the proposed distribution?

b. What is the probability that depth ratio exceeds 2 ?

c. What is the median of the depth ratio distribution?

d. What is the 99th percentile of the depth ratio distribution?

83

What condition on $\alpha$ and $\beta$ is necessary for the standard beta pdf to be symmetric?

84

Suppose the proportion $X$ of surface area in a randomly selected quadrat that is covered by a certain plant has a standard beta distribution with $\alpha =5$ and $\beta =2$ .

a. Compute $E\left(X\right)$ and $V\left(X\right)$ .

b. Compute $P\left(X\le .2\right)$ .

c. Compute $P\left(.2\le X\le .4\right)$ .

d. What is the expected proportion of the sampling region not covered by the plant?

85

Let $X$ have a standard beta density with parameters $\alpha$ and $\beta$ .

a. Verify the formula for $E\left(X\right)$ given in the section.

b. Compute $E\left[{\left(1-X\right)}^m\right]$ . If $X$ represents the proportion of a substance consisting of a particular ingredient, what is the expected proportion that does not consist of this ingredient?

86

Stress is applied to a 20-in. steel bar that is clamped in a fixed position at each end. Let $Y=$ the distance from the left end at which the bar snaps. Suppose $Y/20$ has a standard beta distribution with $E\left(Y\right)=10$ and $V\left(Y\right)=\frac{100}{7}$ .

a. What are the parameters of the relevant standard beta distribution?

b. Compute $P\left(8\le Y\le 12\right)$ .

c. Compute the probability that the bar snaps more than 2 in. from where you expect it to.