Problem 3: (20 points) Consider the integral a = f: f(:12)d:1:. It can be interpreted as the following expectation Elf (U )l, Where U is a random variable that is uniformly distributed on the interval (0, 1). Consider next a sequence U1, U2, - - - , Un of independent and uniformly distributed random variables in the interval (0, 1). 1. (10 points) Show that lim l 2 f(Uz-) = 0., almost surely. z=1 n>+oo n . 2. (20 points) What is the distribution of the error 1 Ti, E;f(Ui)a