Let U1, U2, . . . be independent random variables, each with the uniform distribution on the interval (0, 1). For each integer n 1,

let Mn = min(U1, U2, . . . , Un).

(a) For 0 < x < 1, find P(Mn > x). Explain or calculate.

(b) Find E(Mn). Explain or calculate.

(c) As n gets large, what happens to the distribution of Mn, and why? You are welcome to calculate, but it is also OK to explain qualitatively.

(d) Let Xn = nMn. For large n, approximately what is the distribution of Xn? Prove your answer. It will help to apply a result you established in an earlier part.