Explanation Given info: Consider f ( x ) = θ x θ − 1 , 0 < θ < ∞ and 0 < x < 1 Calculation: Maximum likelihood estimator: If X is a random variable with probability distribution f ( x ; θ ) where θ is an unknown parameter. Consider x 1 , x 2 , ... , x n are the observed values of a random sample of size n. Then the likelihood function of the sample is, L ( θ ) = f ( x 1 ; θ ) ⋅ f ( x 2 ; θ ) ⋅ ⋅ ⋅ f ( x n ; θ ) The likelihood function is a function of only the unknown parameter θ . The maximum likelihood estimator of θ is the value of θ which maximizes the likelihood function L ( θ ) . The likelihood function is, L ( θ ) = ∏ i = 1 n θ x θ − 1 = θ x 1 θ − 1 × θ x 2 θ − 1 × ⋅ ⋅ ⋅ × θ x n θ − 1 = θ n ∏ i = 1 n x i θ − 1 Apply logarithm on both sides. ln L ( θ ) = ln [ θ n ∏ i = 1 n x i θ − 1 ] ln L ( θ ) = ln θ n + ln ( ∏ i = 1 n x i θ<

Applied Statistics And Probability For Engineers 6e Binder Ready Version + Wileyplus Registration Card

6th Edition

ISBN: 9781118865767

Author: Douglas C. Montgomery, George C. Runger

Publisher: WILEY

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Chapter 7, Problem 72SE

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Show that θ^=−n/(ln∏i=1nXi) is the maximum likelihood estimator of θ.

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Chapter 7, Problem 71SE

Chapter 7, Problem 73SE

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Chapter 7 Solutions

Applied Statistics And Probability For Engineers 6e Binder Ready Version + Wileyplus Registration Card

Ch. 7.2 - 7-1. Consider the hospital emergency room data...Ch. 7.2 - 7-2. Consider the compressive strength data in...Ch. 7.2 - 7-3. PVC pipe is manufactured with a mean diameter...Ch. 7.2 - 7-4. Suppose that samples of size n = 25 are...Ch. 7.2 - 7-5. A synthetic fiber used in manufacturing...Ch. 7.2 - 7-6. Consider the synthetic fiber in the previous...Ch. 7.2 - 7-7. The compressive strength of concrete is...Ch. 7.2 - 7-8. Consider the concrete specimens in Exercise...Ch. 7.2 - 7-9. A normal population has mean 100 and variance...Ch. 7.2 - 7-10. Suppose that the random variable X has the...

Ch. 7.2 - 7-11. Suppose that X has a discrete uniform...Ch. 7.2 - 7-12. The amount of time that a customer spends...Ch. 7.2 - 7-13. A random sample of size n1 = 16 is selected...Ch. 7.2 - 7-14. A consumer electronics company is comparing...Ch. 7.2 - 7-15. The elasticity of a polymer is affected by...Ch. 7.2 - 7-16. Scientists at the Hopkins Memorial Forest in...Ch. 7.2 - 7-17. From the data in Exercise 6-21 on the pH of...Ch. 7.2 - 7-18. Researchers in the Hopkins Forest (see...Ch. 7.2 - 7-19. Like hurricanes and earthquakes, geomagnetic...Ch. 7.2 - 7-20. Wayne Collier designed an experiment to...Ch. 7.2 - 7-21. Consider a Weibull distribution with shape...Ch. 7.3 - 7-22. A computer software package calculated some...Ch. 7.3 - 7-23. A computer software package calculated some...Ch. 7.3 - 7-24. Let X1 and X2 be independent random...Ch. 7.3 - 7-25. Suppose that we have a random sample X1,...Ch. 7.3 - 7-26. Suppose we have a random sample of size 2n...Ch. 7.3 - 7-27. Let X1 , X2 ,…, X7 denote a random sample...Ch. 7.3 - 7-28. Suppose that and are unbiased estimators...Ch. 7.3 - 7-29. Suppose that and are estimators of the...Ch. 7.3 - 7-30. Suppose that are estimators of θ. We know...Ch. 7.3 - 7-31. Let three random samples of sizes n1 = 20,...Ch. 7.3 - 7-32. (a) Show that is a biased estimator of...Ch. 7.3 - 7-33. Let X1 ,X2, … ,Xn be a random sample of size...Ch. 7.3 - 7-34. Data on pull-off force (pounds) for...Ch. 7.3 - 7-35. Data on the oxide thickness of semiconductor...Ch. 7.3 - 7-36. Suppose that X is the number of observed...Ch. 7.3 - 7-37. and are the sample mean and sample...Ch. 7.3 - 7-38. Two different plasma etchers in a...Ch. 7.3 - 7-39. Of n1 randomly selected engineering students...Ch. 7.4 - 7-44. Let X be a geometric random variable with...Ch. 7.4 - 7-45. Consider the Poisson distribution with...Ch. 7.4 - 7-46. Let X be a random variable with the...Ch. 7.4 - 7-48. Consider the probability density...Ch. 7.4 - 7-49. Let X1, X2, … Xn be uniformly distributed on...Ch. 7.4 - 7-50. Consider the probability density...Ch. 7.4 - 7-51. The Rayleigh distribution has probability...Ch. 7.4 - 7-52. Let X1, X2, …, Xn be uniformly distributed...Ch. 7.4 - 7-53. Consider the Weibull distribution (a) Find...Ch. 7.4 - 7-55. Suppose that X is a normal random variable...Ch. 7.4 - 7-56. Suppose that X is a normal random variable...Ch. 7.4 - 7-57. Suppose that X is a Poisson random variable...Ch. 7.4 - 7-58. Suppose that X is a normal random variable...Ch. 7.4 - 7-59. The weight of boxes of candy is a normal...Ch. 7.4 - 7-60. The time between failures of a machine has...Ch. 7 - Prob. 61SE Ch. 7 - 7-62. Suppose that a random variable is normally...Ch. 7 - Prob. 63SE Ch. 7 - 7-64. A procurement specialist has purchased 25...Ch. 7 - 7-65. A random sample of 36 observations has been...Ch. 7 - Prob. 66SE Ch. 7 - Prob. 67SE Ch. 7 - Prob. 68SE Ch. 7 - 7-69. A manufacturer of semiconductor devices...Ch. 7 - Prob. 70SE Ch. 7 - Prob. 71SE Ch. 7 - Prob. 72SE Ch. 7 - Prob. 73SE Ch. 7 - 7-74. You plan to use a rod to lay out a square,...Ch. 7 - Prob. 75SE Ch. 7 - Prob. 76SE Ch. 7 - Prob. 77SE Ch. 7 - Prob. 78SE Ch. 7 - Prob. 79SE Ch. 7 - Prob. 80SE Ch. 7 - Prob. 81SE Ch. 7 - 7-82. Let X be a random variable with mean μ and...Ch. 7 - Prob. 83SE Ch. 7 - Prob. 84SE

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