A continuous random variable X has probability density function k/x³, 1 ≤ x ≤ 2, fx (x) = 0, elsewhere, where k is an appropriate constant. (a) Calculate the value of k. (b) Find the expectation and variance of X. (c) Find the cumulative distribution function Fx (x) and hence calculate the probabilities Pr(X<4/3) and Pr(X² < 2). (d) Let X1, X2, X3, . be a sequence of random variables distributed as the random variable X. In our case, which conditions of the central limit theorem are satisfied? Do we need any other assumptions? Explain your answer. (e) Let Y = X² - 1. Find the density function of Y.
