(R) The Central Limit Theorem (a) Consider a continuous random variable Xi Uni f orm [0, 2]. What is E [Xi ] and Var (Xi)? (b) Consider the Xi defined above in (a) for i = 1, 2, ..., n, where each Xi Xj whenever i = j. Consider the sample mean X n := 1 n n i=1 Xi . What is E [X n] and Var (X n)? (The answer would be a function of n). (c) Consider the transformation Yn := n (X n E [Xi ]). What is E [Yn] and Var (Yn)? (d) Consider the transformation Zn := n (X n E [Xi ]) p Var (Xi) . What is E [Zn] and Var (Zn)? Now repeat the following for n = 1, 2, 3, 5, 10, 50, 100, 1000, 3000. Use the for loops to execute (e)-(m). (e) (R) Generate t = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ..., 2500 n 1 vectors of independent Uni f orm [0, 2] random variables and calculate its sample mean X t n respectively for each t. Denote this size