Applied Statistics and Probability for Engineers 5th Ed.

["","","","","","","","","","","","","","","7-4 METHODS OF POINT ESTIMATION 247 n 1\u0580\u2424 machines are observed, and the average time between failures xi\u2424 is x \u03ed 1125 hours. \u00a3 a \u00a7 (a) Find the Bayes estimate for \u242d. \u2426\u03ed (b) What proportion of the machines do you think will fail i\u03ed1 before 1000 hours? n Supplemental Exercises (c) What complications are involved in solving the two equa- 7-51. Transistors have a life that is exponentially distributed tions in part (b)? with parameter \u242d. A random sample of n transistors is taken. What is the joint probability density function of the sample? 7-44. Reconsider the oxide thickness data in Exercise 7-29 7-52. Suppose that a random variable is normally distrib- and suppose that it is reasonable to assume that oxide thick- uted with mean \u242e and variance \u24342, and we draw a random sample of \ufb01ve observations from this distribution. What is the ness is normally distributed. joint probability density function of the sample? (a) Compute the maximum likelihood estimates of \u242e and \u24342. (b) Graph the likelihood function in the vicinity of \u242e\u02c6 and \u2434\u02c6 2, 7-53. Suppose that X is uniformly distributed on the interval from 0 to 1. Consider a random sample of size 4 from X. What the maximum likelihood estimates, and comment on its is the joint probability density function of the sample? shape. 7-54. A procurement specialist has purchased 25 resistors (c) Suppose that the sample size was larger (n \u03ed 40) but the from vendor 1 and 30 resistors from vendor 2. Let X1,1, X1,2, p , X1,25 represent the vendor 1 observed resistances, maximum likelihood estimates were numerically equal to which are assumed to be normally and independently distrib- uted with mean 100 ohms and standard deviation 1.5 ohms. the values obtained in part (a). Graph the likelihood func- Similarly, let X2,1, X2,2, p , X2,30 represent the vendor 2 ob- tion for n \u03ed 40, compare it to the one from part (b), and served resistances, which are assumed to be normally and in- comment on the effect of the larger sample size. dependently distributed with mean 105 ohms and standard deviation of 2.0 ohms. What is the sampling distribution of 7-45. Suppose that X is a normal random variable with un- X1 \u03ea X2 ? What is the standard error of X1 \u03ea X2? known mean \u242e and known variance \u24342. The prior distribution 7-55. A random sample of 36 observations has been drawn for \u242e is a normal distribution with mean \u242e0 and variance \u243402 . from a normal distribution with mean 50 and standard deviation Show that the Bayes estimator for \u242e becomes the maximum 12. Find the probability that the sample mean is in the interval likelihood estimator when the sample size n is large. 47 \u0545 X \u0545 53. Is the assumption of normality important? Why? 7-56. A random sample of n \u03ed 9 structural elements is 7-46. Suppose that X is a normal random variable with un- tested for compressive strength. We know that the true mean known mean \u242e and known variance \u24342. The prior distribution compressive strength \u242e \u03ed 5500 psi and the standard deviation for \u242e is a uniform distribution de\ufb01ned over the interval [a, b]. is \u2434 \u03ed 100 psi. Find the probability that the sample mean (a) Find the posterior distribution for \u242e. compressive strength exceeds 4985 psi. (b) Find the Bayes estimator for \u242e. 7-57. A normal population has a known mean 50 and 7-47. Suppose that X is a Poisson random variable with pa- known variance \u24342 \u03ed 2. A random sample of n \u03ed 16 is se- rameter \u242d. Let the prior distribution for \u242d be a gamma distri- lected from this population, and the sample mean is x \u03ed 52. How unusual is this result? bution with parameters m \u03e9 1 and 1m \u03e9 12\u0580\u242d0. 7-58. A random sample of size n \u03ed 16 is taken from a nor- (a) Find the posterior distribution for \u242d. mal population with \u242e \u03ed 40 and \u24342 \u03ed 5. Find the probability (b) Find the Bayes estimator for \u242d. that the sample mean is less than or equal to 37. 7-48. Suppose that X is a normal random variable with 7-59. A manufacturer of semiconductor devices takes a unknown mean and known variance \u24342 \u03ed 9. The prior distri- random sample of 100 chips and tests them, classifying each bution for \u242e is normal with \u242e0 \u03ed 4 and \u243420 \u03ed 1. A random chip as defective or nondefective. Let Xi \u03ed 0 if the chip is sample of n \u03ed 25 observations is taken, and the sample mean nondefective and Xi \u03ed 1 if the chip is defective. The sample is x \u03ed 4.85. fraction defective is (a) Find the Bayes estimate of \u242e. (b) Compare the Bayes estimate with the maximum likeli- P\u02c6 \u03ed X1 \u03e9 X2 \u03e9 p \u03e9 X100 100 hood estimate. What is the sampling distribution of the random variable P\u02c6? 7-49. The weight of boxes of candy is a normal random variable with mean \u242e and variance 1\u058010 pound. The prior dis- tribution for \u242e is normal, with mean 5.03 pound and variance 1\u058025 pound. A random sample of 10 boxes gives a sample mean of x \u03ed 5.05 pounds. (a) Find the Bayes estimate of \u242e. (b) Compare the Bayes estimate with the maximum likeli- hood estimate. 7-50. The time between failures of a machine has an expo- nential distribution with parameter \u242d. Suppose that the prior distribution for \u242d is exponential with mean 100 hours. Two","248 CHAPTER 7 SAMPLING DISTRIBUTIONS AND POINT ESTIMATION OF PARAMETERS 7-60. Let X be a random variable with mean \u242e and variance 7-64. You plan to use a rod to lay out a square, each side of \u24342. Given two independent random samples of sizes n1 and n2, which is the length of the rod. The length of the rod is \u242e, which with sample means X1 and X2, show that is unknown. You are interested in estimating the area of the square, which is \u242e2. Because \u242e is unknown, you measure it n X \u03ed aX1 \u03e9 11 \u03ea a2 X2, 0 \u03fd a \u03fd 1 times, obtaining observations X1, X2, p , Xn. Suppose that each measurement is unbiased for \u242e with variance \u24342. is an unbiased estimator for \u242e. If X1 and X2 are independent, (a) Show that X2 is a biased estimate of the area of the square. \ufb01nd the value of a that minimizes the standard error of X . (b) Suggest an estimator that is unbiased. 7-61. A random variable x has probability density function 7-65. An electric utility has placed special meters on 10 houses in a subdivision that measures the energy consumed f 1x2 \u03ed 1 x2e\u03eax\u0580\u242a, 0 \u03fd x \u03fd \u03f1, 0 \u03fd \u242a \u03fd \u03f1 (demand) at each hour of the day. They are interested in the en- 2\u242a3 ergy demand at one speci\ufb01c hour, the hour at which the system experiences the peak consumption. The data from these 10 me- Find the maximum likelihood estimator for \u242a. ters are as follows (in KW): 23.1, 15.6, 17.4, 20.1, 19.8, 26.4, 25.1, 20.5, 21.9, and 28.7. If \u242e is the true mean peak demand 7-62. Let f 1x2 \u03ed \u242ax\u242a\u03ea1, 0 \u03fd \u242a \u03fd \u03f1, and 0 \u03fd x \u03fd 1. for the ten houses in this group of houses having the special n meters, estimate \u242e. Now suppose that the utility wants to esti- Show that \u2330\u02c6 \u03ed \u03ean\u0580 1ln w i\u03ed1 Xi2 is the maximum likelihood mate the demand at the peak hour for all 5,000 houses in this subdivision. Let \u242a be this quantity. Estimate \u242a using the data estimator for \u242a. given above. Estimate the proportion of houses in the subdivi- sion that demand at least 20KW at the hour of system peak. 7-63. Let f 1x2 \u03ed 11\u0580\u242a2x 11\u03ea\u242a2\u0580\u242a, 0 \u03fd x \u03fd 1, and 0 \u03fd \u242a \u03fd \u03f1. \u2330\u02c6 \u03ea11\u0580n2 n Show that \u03ed g i\u03ed 1 ln1Xi2 is the maximum likelihood estimator for \u242a and that \u2330\u02c6 is an unbiased estimator for \u242a. MIND-EXPANDING EXERCISES 7-66. A lot consists of N transistors, and of these, M (b) Find the value of cn for n \u03ed 10 and n \u03ed 25. (M \u0545 N) are defective. We randomly select two transis- Generally, how well does S perform as an estimator tors without replacement from this lot and determine whether they are defective or nondefective. The of \u2434 for large n with respect to bias? random variable 7-68. A collection of n randomly selected parts is 1, if the ith transistor measured twice by an operator using a gauge. Let Xi and Xi \u03ed \u03bc 0, is nondefective Yi denote the measured values for the ith part. Assume if the ith transistor that these two random variables are independent and is defective i \u03ed 1, 2 normally distributed and that both have true mean \u242ei and variance \u24342. (a) Show that the maximum likelihood estimator of \u24342 n is \u2434\u02c6 2 \u03ed 11\u05804n2 g i\u03ed1 1Xi \u03ea Yi22. Determine the joint probability function for X1 and X2. (b) Show that \u2434\u02c6 2 is a biased estimator for \u2434\u02c6 2. What What are the marginal probability functions for X1 and X2? Are X1 and X2 independent random variables? happens to the bias as n becomes large? (c) Find an unbiased estimator for \u24342. 7-67. When the sample standard deviation is based on a random sample of size n from a normal population, it can 7-69. Consistent Estimator. Another way to measure be shown that S is a biased estimator for \u2434. Speci\ufb01cally, the closeness of an estimator \u235c\u02c6 to the parameter \u242a is in terms of consistency. If \u235c\u02c6 n is an estimator of \u242a based on a random sample of n observations, \u235c\u02c6 n is consistent for \u242a if E1S 2 \u03ed \u243412\u05801n \u03ea 12 \u232b1n\u058022\u0580 \u232b3 1n \u03ea 12\u058024 lim P 1 0\u235c\u02c6 n \u03ea \u242a0 \u03fd \u24402 \u03ed 1 (a) Use this result to obtain an unbiased estimator for \u2434 nS\u03f1 of the form cnS, when the constant cn depends on the sample size n. Thus, consistency is a large-sample property, describing the limiting behavior of \u235c\u02c6 n as n tends to in\ufb01nity. It is usually dif\ufb01cult to prove consistency using the above","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","",""]