a. Use Equation (4.13) to write a formula for the median of the lognormal distribution....

a. Use Equation (4.13) to write a formula for the median of the lognormal distribution. What is the median for the load distribution of Exercise 79?

b. Recalling that z_α 100(1 – α)is our notation for the percentile of the standard normal distribution, write an expression for the 100(1 – α) percentile of the lognormal distribution. In Exercise 79, what value will load exceed only 1% of the time?

Reference 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 = .40(cv = σ_x/μ_x).

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?

Reference equation 4.13