1. (20 points) Let {X} be a sequence of iid random variables with common pdf f(x) = 1 -12/2 VER. √2πT Then find the limit in probability of the sequence of random variables 1 {Y} where Y₁ = ΣΙΧ). n i=1 iid 2. (20 points) Let {X} Uniform(0, 1). Define Y = min{X1,..., Xn}, Zn = max{X1,..., Xn}, Un=nYn, Vn = n(1 - Z Establish the following convergence in distribution. HINT: Apply the definition vergence in distribution directly and work with cdfs of order statistics. (a) Un > Exp(1). (b) Vn>Exp(1). 3. (20 points) Let {X} be iid with E[X₁] = 0 and E[X2] = 2. Find the limit in d