Chapter 5.1, Problem 11E

Statisticians frequently use the extreme extreme value distribution given by the cdf

$F\left(x\right)=1-\exp \left[-e^{\frac{\left(x-{\theta }_1\right)}{{\theta }_2}}\right],-\infty <x<\infty .$ A simple case is when ${\theta }_1=0$ and ${\theta }_2=1$, giving

$F\left(x\right)=1-\exp \left[-e^x\right],-\infty <x<\infty .$

Let $Y=e^X$ or $X=\ln Y$: then the support of Y is $0<y<\infty .$

(a) Show that the distribution of Y is exponential when ${\theta }_1=0$ and ${\theta }_2=1$.

(b) Find the cdf and the pdf of Y when ${\theta }_1\ne 0$ and ${\theta }_2>1$.

(c) Let ${\theta }_1=\ln \beta$ and ${\theta }_2=\frac{1}{\alpha }$ in the cdf and pdf of Y. What is this distribution?

(d) As suggested by its name, the extreme value distribution can be used to model the longest home run, the deepest mine, the greatest flood, and so on. Suppose the length X (in feet) of the maximum of someone’s home runs was modeled by an extreme value distribution with ${\theta }_1=550$ and ${\theta }_2=25$.

What is the probability that X exceeds 500 feet?