Question 2 [14 marks] (a) A random variable X has probability density function x) for O < x < 1, {0 otherwise. (i) For this distribution, show that p = E(X) = } , = var(X) (ii) Show that Var(X2) and E(X') (iii) Suppose that XL, , Xn is a random sample from the distribution of X as spec- ified above. Show that as n Y 10 (b) Suppose a production line has the probability of p = 0.02 to produce faulty items. If this production line produces a fixed number of 300 items per day, use the Central Limit Theorem to approximately calculate the probability that the number of faulty items produced by this production line in a particular month (30 days) is greater than 300. (You may express your answer in terms of the standard Normal cumulative distri bution function without obtaining the final result.)