a. Use Equation (4.13) to write a formula for the median ,m of the lognormal distribution. What is the median for the load distribution of Exercise 79?
b. Recalling that za is our notation for the 100(1 -α ) percentile of the standard normal distribution, write an expression for the 100(1 2 a) percentile of the lognormal distribution. In Exercise 79, what value will load exceed only 1% of the time?
Exercise 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 kg/day/km and a coefficient of variation CV 5 .40 (CV = sXymX).
a. What are the mean value and standard deviation of ln(X)?
b. What is the probability that X is at most 15,000 kg/day/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?