Let X1, X2, . . . be independent, U (0, 1)-distributed random variables, and let Nm ∈ Po(m) be independent of X1, X2, . . . . Set Vm = max{X1, . . . , XNm } (Vm = 0 when Nm = 0). Determine

(a) the distribution function of Vm,

(b) the moment generating function of Vm.

It is reasonable to believe that Vm is "close" to 1 when m is "large" (cf. Problem 8.1). The purpose of parts (c) and (d) is to show how this can be made more precise.

(c) Show that E Vm → 1 as m → ∞.

(d) Show that m(1 - Vm) converges in distribution as m → ∞, and determine the limit distribution.