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# Mathematical Statistics with Applications, Seventh Edition

www.downloadslide.com Supplementary Exercises 341 6.8 Summary This chapter has been concerned with finding probability distributions for functions of random variables. This is an important problem in statistics because estimators of population parameters are functions of random variables. Hence, it is necessary to know something about the probability distributions of these functions (or estimators) in order to evaluate the goodness of our statistical procedures. A discussion of estimation will be presented in Chapters 8 and 9. The methods for finding probability distributions for functions of random variables are the distribution function method (Section 6.3), the transformation method (Section 6.4), and the moment-generating-function method (Section 6.5). It should be noted that no particular method is best for all situations because the method of solution depends a great deal upon the nature of the function involved. If U 1 and U 2 are two functions of the continuous random variables Y 1 and Y 2 , the joint density function for U 1 and U 2 can be found using the Jacobian technique in Section 6.6. Facility for handling these methods can be achieved only through practice. The exercises at the end of each section and at the end of the chapter provide a good starting point. The density functions of order statistics were presented in Section 6.7. Some special functions of random variables that are particularly useful in statistical inference will be considered in Chapter 7. References and Further Readings Casella, G., and R. L. Berger. 2002. Statistical Inference, 2d ed. Pacific Grove, Calif.: Duxbury. Hoel, P. G. 1984. Introduction to Mathematical Statistics, 5th ed. New York: Wiley. Hogg, R. V., A. T. Craig, and J. W. McKean. 2005. Introduction to Mathematical Statistics, 6th ed. Upper Saddle River, N.J.: Pearson Prentice Hall. Mood, A. M., F. A. Graybill, and D. Boes. 1974. Introduction to the Theory of Statistics, 3d ed. New York: McGraw-Hill. Parzen, E. 1992. Modern Probability Theory and Its Applications. New York: Wiley-Interscience. Supplementary Exercises 6.92 If Y 1 and Y 2 are independent and identically distributed normal random variables with mean µ and variance σ 2 , find the probability density function for U = (1/2)(Y 1 − 3Y 2 ). 6.93 When current I flows through resistance R, the power generated is given by W = I 2 R. Suppose that I has a uniform distribution over the interval (0, 1) and R has a density function given by { 2r, 0 ≤ r ≤ 1, f (r) = 0, elsewhere. Find the probability density function for W . (Assume that I is independent of R.)

342 Chapter 6 Functions of Random Variables www.downloadslide.com 6.94 Two efficiency experts take independent measurements Y 1 and Y 2 on the length of time workers take to complete a certain task. Each measurement is assumed to have the density function given by { (1/4)ye −y/2 , y > 0, f (y) = 0, elsewhere. Find the density function for the average U = (1/2)(Y 1 + Y 2 ).[Hint: Use the method of moment-generating functions.] 6.95 Let Y 1 and Y 2 be independent and uniformly distributed over the interval (0, 1). Find the probability density function of each of the following: a U 1 = Y 1 /Y 2 . b U 2 =−ln (Y 1 Y 2 ). c U 3 = Y 1 Y 2 . 6.96 Suppose that Y 1 is normally distributed with mean 5 and variance 1 and Y 2 is normally distributed with mean 4 and variance 3. If Y 1 and Y 2 are independent, what is P(Y 1 > Y 2 )? 6.97 Suppose that Y 1 is a binomial random variable with four trials and success probability .2 and that Y 2 is an independent binomial random variable with three trials and success probability .5. Let W = Y 1 + Y 2 . According to Exercise 6.53(e), W does not have a binomial distribution. Find the probability mass function for W .[Hint: P(W = 0) = P(Y 1 = 0, Y 2 = 0); P(W = 1) = P(Y 1 = 1, Y 2 = 0) + P(Y 1 = 0, Y 2 = 1); etc.] *6.98 The length of time that a machine operates without failure is denoted by Y 1 and the length of time to repair a failure is denoted by Y 2 . After a repair is made, the machine is assumed to operate like a new machine. Y 1 and Y 2 are independent and each has the density function { e −y , y > 0, f (y) = 0, elsewhere. Find the probability density function for U = Y 1 /(Y 1 + Y 2 ), the proportion of time that the machine is in operation during any one operation–repair cycle. *6.99 Refer to Exercise 6.98. Show that U, the proportion of time that the machine is operating during any one operation–repair cycle, is independent of Y 1 + Y 2 , the length of the cycle. 6.100 The time until failure of an electronic device has an exponential distribution with mean 15 months. If a random sample of five such devices are tested, what is the probability that the first failure among the five devices occurs a b after 9 months? before 12 months? *6.101 A parachutist wants to land at a target T , but she finds that she is equally likely to land at any point on a straight line (A, B), of which T is the midpoint. Find the probability density function of the distance between her landing point and the target. [Hint: Denote A by −1, B by +1, and T by 0. Then the parachutist’s landing point has a coordinate X, which is uniformly distributed between −1 and +1. The distance between X and T is |X|.] 6.102 Two sentries are sent to patrol a road 1 mile long. The sentries are sent to points chosen independently and at random along the road. Find the probability that the sentries will be less than 1/2 mile apart when they reach their assigned posts. *6.103 Let Y 1 and Y 2 be independent, standard normal random variables. Find the probability density function of U = Y 1 /Y 2 .

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