7.10 Consider {Yn}n≥1 a sequence of independent identically distributed random variables with common distribution U([0, 1]) (uniform on the interval). For every n ≥ 1, we define Yn

(1) = min(Y1,...,Yn)

and Yn

(n) = max(Y1,...,Yn).

the first and last order statistics of the sequence, respectively. Let Fn(x), Gn(x) denote respectively the cdfs of these random variables.

a) Show that for every x the sequence Fn(x) converges, and find the limit. Use this result to conclude that the sequence of random variables Yn

(1) converges in distribution to 0.

b) Show that for every x the sequence Gn(x) converges, and find the limit. Use this result to conclude that the sequence of random variables Yn

(n) converges in distribution to 1.