EX 4.27

According to the article “Predictive Model for Pitting Corrosion in Buried Oil and Gas Pipelines” (Corrosion, 2009: 332-342), the lognormal distribution has been reported as the best option for describing the distribution of maximum pit depth data from cast iron pipes in soil. The authors suggest that a lognormal distribution with $\mu =.353$ and $\sigma =.754$ is appropriate for maximum pit depth $\left(\mathrm{mm}\right)$ of buried pipelines. For this distribution, the mean value and variance of pit depth are

$$E\left(X\right)=e^{.353+{\left(.754\right)}^2/2}=e^{.6373}=1.891$$ $$V\left(X\right)=e^{2\left(.353\right)+{\left(.754\right)}^2}\cdot \left(e^{{\left(.754\right)}^2}-1\right)=\left(3.57697\right)\left(.765645\right)=2.7387$$

The probability that maximum pit depth is between 1 and $2\mathrm{mm}$ is

$$\begin{array}{rl}& P\left(1\le X\le 2\right)\\ & =P\left(\ln \left(1\right)\le \ln \left(X\right)\le \ln \left(2\right)\right)\\ & =P\left(0\le \ln \left(X\right)\le 0.693\right)\\ & =P\left(\frac{0-0.353}{0.754}\le Z\le \frac{0.693-0.353}{0.754}\right)\\ & =\Phi \left(0.47\right)-\Phi \left(-0.45\right)\\ & =0.354\end{array}$$

This probability is illustrated in Figure 4.31 (from Minitab).

Figure 4.31 Lognormal density curve with $\mu =.353$ and $\sigma =.754$

What value $c$ is such that only $1\%$ of all specimens have a maximum pit depth exceeding $c$ ? The desired value satisfies

$$.99=P\left(X\le c\right)=P\left(Z\le \frac{\ln \left(c\right)-.353}{.754}\right)$$

The $z$ critical value 2.33 captures an upper-tail area of $.01\left(z_{.01}=2.33\right)$ , and thus a cumulative area of .99 . This implies that

$$\frac{\ln \left(c\right)-.353}{.754}=2.33$$

from which $\ln \left(c\right)=2.1098$ and $c=8.247$ . Thus 8.247 is the 99th percentile of the maximum pit depth distribution.