Let X1,...,Xn be n independent, exponentially distributed waiting times with common intensity , and define Mn =

Let X1,...,Xn be n independent, exponentially distributed waiting times with common intensity λ, and define Mn = max1≤i≤n Xi. Show that λMn − ln n converges in distribution to the extreme value statistic having density e−e−u e−u. (Hints: This assertion can be most easily demonstrated by considering the moment generating function of λMn−ln n. Since Mn has density n(1−e−λx)n−1λe−λx, prove that E[es(λMn−ln n)

] =  ∞

0 es(λx−ln n)

n(1 − e−λx)

n−1λe−λxdx

=

 ∞

− ln n esu(1 − 1 n

e−u)

n−1e−udu.

Argue that in the limit limn→∞ E[es(λMn−ln n)

] =  ∞

−∞

esue−e−u e−udu

=

 ∞

0 w−se−wdw (14.9)

= Γ(1 − s), where Γ(x) is the gamma function. )