The article "The Prediction of Corrosion by Statistical Analysis of Corrosion Profiles" (Corrosion Science, 1985: 305-315) suggests the following cdf for the depth X of the deepest pit in an experiment involving the exposure of carbon manganese steel to acidified seawater.

F (x; α, β) = e -e -(x- α) / B -∞ < x < ∞

The authors propose the values a 5 150 and b 5 90.

Assume this to be the correct model.

a. What is the probability that the depth of the deepest pit is at most 150? At most 300? Between 150 and 300?

b. Below what value will the depth of the maximum pit be observed in 90% of all such experiments?

c. What is the density function of X?

d. The density function can be shown to be unimodal (a single peak). Above what value on the measurement axis does this peak occur? (This value is the mode.)

e. It can be shown that E(X) ≈ .5772 β + α. What is the mean for the given values of α and β, and how does it compare to the median and mode? Sketch the graph of the density function. [This is called the largest extreme value distribution.]