(a) Use expression (3.4) to show that if tn) is the largest of η independent unit exponential…

(a) Use expression (3.4) to show that if tn) is the largest of η independent unit exponential variates, the rth cumulant of I t(n )is

(b) Show that the asymptotic distribution of is the extreme value distribution with density (2.1). Note also that the MGF of X„ converges uniformly for   , 1) to the MGF of (2.1). This implies

(c) By making use of parts (a) and (b), show that the extreme value density has cumulants