7.31 Order statistics. Suppose X1,...,X100 are independent and uniformly distributed on (0,1). (a) Find the probability the 25th smallest variable is less than 0.20. (b) Find E[X(95)] and V [X(95)]. (c) The range of a set of numbers is the difference between the maximum and the minimum. Find the expected range. 7.34 Find the variance of a beta distribution with parameters a and b. 7.35 Let X  Beta(a, b). Find the distribution of Y = 1  X 7.38 In a population, suppose personal income above $15,000 has a Pareto distribution with a = 1.8 (units are $10,000). Find the probability that a randomly chosen individual has income greater than $60,000. 7.39 In Newman (2005), the population of U.S. cities larger than 40,000 is modeled with a Pareto distribution with a = 2.30. Find the probability that a random city's population is greater than k = 3, 4, 5, and 6 standard deviations above the mean. 7.41 Let X  Exp(a). Let Y = meX. Show that Y  Pareto(m, a). 7.43 The "99-10" rule on the Internet says that 99% of the content generated in Internet chat rooms is created by 10% of the users. If the amount of chat room content has a Pareto distribution, find the value of the parameter a.