2. (10 points) For any n = N, let X1, X2, ..., Xn be independent random variables each of which are uniformly distributed over (0,1), i.e. Xi U(0, 1) for each i = {1, 2, ..., n}. Define the ~ random variable Yn as follows Yn = n min{X1, X2, ..., Xn} (note the prefactor n !!!) . a) For each nЄ N, find the cumulative distribution function and the density of Yn. (Hint: in order to find P(Y, ≤ x), consider the counterevent.) Make sure to write down the distribution and density of Yn for every x = R, so also where they are equal to zero, etc. b) For each nЄ N, compute E(Y). c) Show that for any x € (0,∞), we have lim P(Yn > x) = e¯ . N→X What kind of random variable is Y therefore approximating? (1)