29. Let I = f e2 dx. (a) Show that for any u and o 1 8. V2Ta / e-(x-120² dx = 1 is equivalent to I = 27. (b) Show that I = /2n by writing -312 %3D e-(x²+y³)/2 dy 3 dx dy 212 dx 0- and then evaluating the double integral by means of a change of variables to polar coordinates. (That is, let x=r cos 0, y = r sin 0, dx dy = r dr de 30. A random variable X is said to have a lognormal distribution if log X is normally distributed. If X is lognormal with E[log X] = µ and Var(log X) = o², deter- mine the distribution function of X. That is, what is P{X < x}? 31. The salaries of pediatric physicians are approximately normally distributed. If 25 percent of these physicians earn below 180, 000 and 25 percent earn above 320, 000, what fraction earn (a) below 250, 000; (b) between 260, 00 and 300, 000? 32. The sample mean and sample standard deviation on your economics examina- tion were 60 and 20, respectively; the sample mean and sample standard devia- tion on your statistics examination were 55 and 10, respectively. You scored 70 on the economics exam and 62 on the statistics exam. Assuming that the two histograms of test scores are approximately normal histograms, (a) on which exam was your percentile score highest? (b) approximate the percentage of the scores on the economics exam that were below your sCore. (c) approximate the percentage of the scores on the statistics exam that were below your score. 33. Value at risk (VAR) has become a key concept in financial calculations. The VAR an investment is defined as that value v such that there is only a 1 percent chance that the loss from the investment will exceed v. of (a) If the gain from an investment is a normal random variable with mean and variance 49, determine the value at risk. (If X is the gain, then -X is the loss.) (b) Among a set of investments whose gains are all normally distributed show that the one having the smallest VAR is the one having the largest val ra mee with the groding system for Problems 203 u - 2.330, where u and o are the mean and variance of the gain from the .2 investment. 34. The annual rainfall in Cincinnati is normally distributed with mean 40.14 inches and standard deviation 8.7 inches. (a) What is the probability this year's rainfall will exceed 42 inches? (b) What is the probability that the sum of the next 2 years' rainfall will exceed 84 inches? (c) What is the probability that the sum of the next 3 years' rainfall will exceed 126 inches? (d) For parts (b) and (c), what independence assumptions are you making? 35. The height of adult women in the United States is normally distributed with mean 64.5 inches and standard deviation 2.4 inches. Find the probability that a randomly chosen woman is (a) less than 63 inches tall; (b) less than 70 inches tall; (c) between 63 and 70 inches tall. (d) Alice is 72 inches tall. What percentage of women is shorter than Alice? (e) Find the probability that the average of the heights of two randomly chosen women exceeds 66 inches. (f) Repeat part (e) for four randomly chosen women. 36. An IQ test produces scores that are normally distributed with mean value 100 and standard deviation 14.2. The top 1 percent of all scores are in what range? 37. The time (in hours) required to repair a machine is an exponentially distributed random variable with parameter A = 1. %3D (a) What is the probability that a repair time exceeds 2 hours? (b) What is the conditional probability that a repair takes at least 3 hours, given that its duration exceeds 2 hours? 38. The number of years a radio functions is exponentially distributed with 3: If Jones buys a used radio, what is the probability that it will be param- eter 2 working after an additional 10 years? 39. Jones figures that the total number of thousands of miles that a used auto can be driven before it would need to be junked is an exponential random variable with parameter 20. Smith has a used car that he claims has been driven only 10,000 miles. If Jones purchases the car, what is the probability that she would get at least 20,000 additional miles out of it? Repeat under the assumption that the lifetime mileage of the car is not exponentially distributed but rather is (in thousands of miles) uniformly distributed over (0, 40).