Explanation Calculation: It is given that X and Y are independent continuous uniform random variables over the region 0 < x < 1 and 0 < y < 1 . Uniform distribution: The random variable X is said to follow continuous uniform distribution, if the probability density function is, f X ( x ) = 1 b − a for a < x < b . The joint probability density function of the continuous random variable X and Y is defined as f XY ( x , y ) . The properties of joint probability density function is, f XY ( x , y ) ≥ 0 ∫ − ∞ ∞ ∫ − ∞ ∞ f XY ( x , y ) d x d y = 1 For any region R of two-dimensional space, P ( ( X , Y ) ∈ R ) = ∬ R f X Y ( x , y ) d x d y As f ( x , y ) satisfy the properties of joint probability density function then ∫ − ∞ ∞ ∫ − ∞ ∞ f XY ( x , y ) = 1 . The range of X and Y are 0 < x < 1 and 0 < y < 1 . Thus, ∫ − ∞ ∞ ∫ − ∞ ∞ f XY ( x , y ) d x d y = 1 ∫ 0 1 ∫ 0 1 c d x d y = 1 c ∫ 0 1 d x ∫ 0 1 d y = 1 c [ x ] 0 1 [ y ] 0 1 = 1 c ( 1 − 0 ) ( 1 − 0 ) = 1 c ( 1 ) ( 1 ) = 1 c × 1 = 1 c = 1 Hence, c = 1

Applied Statistics and Probability for Engineers

6th Edition

ISBN: 9781118539712

Author: Douglas C. Montgomery

Publisher: WILEY

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Chapter 5, Problem 107SE

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Chapter 5, Problem 106SE

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Chapter 5 Solutions

Applied Statistics and Probability for Engineers

Ch. 5.1 - Prob. 1E Ch. 5.1 - Prob. 2E Ch. 5.1 - Prob. 3E Ch. 5.1 - 5-4. Four electronic printers are selected from a...Ch. 5.1 - 5-5. In the transmission of digital information,...Ch. 5.1 - 5-6. A small-business Web site contains 100 pages...Ch. 5.1 - 5-7. A manufacturing company employs two devices...Ch. 5.1 - 5-8. Suppose that the random variables X, Y, and Z...Ch. 5.1 - 5-9. An engineering statistics class has 40...Ch. 5.1 - 5-10. An article in the Journal of Database...

Ch. 5.1 - 5-11. For the Transaction Processing Performance...Ch. 5.1 - 5-12. In the transmission of digital information,...Ch. 5.1 - Prob. 13E Ch. 5.1 - Prob. 14E Ch. 5.1 - Prob. 15E Ch. 5.1 - Prob. 16E Ch. 5.1 - Prob. 17E Ch. 5.1 - 5-18. The conditional probability distribution of...Ch. 5.1 - Prob. 19E Ch. 5.1 - 5-20. The time between surface finish problems in...Ch. 5.1 - 5-21. A popular clothing manufacturer receives...Ch. 5.1 - 5-22. The blade and the bearings are important...Ch. 5.1 - Prob. 23E Ch. 5.1 - Prob. 24E Ch. 5.1 - 5-25. Determine the value of c that makes fXYZ (x,...Ch. 5.1 - 5-26. The yield in pounds from a day’s production...Ch. 5.1 - 5-27. The weights of adobe bricks used for...Ch. 5.1 - 5-28. A manufacturer of electroluminescent lamps...Ch. 5.1 - 5-29. The lengths of the minor and major axes are...Ch. 5.1 - 5-30. An article in Health Economics [“Estimation...Ch. 5.1 - 5-31. An article in Clinical Infectious Diseases...Ch. 5.1 - Prob. 32E Ch. 5.2 - Prob. 33E Ch. 5.2 - Prob. 34E Ch. 5.2 - Prob. 35E Ch. 5.2 - Prob. 36E Ch. 5.2 - Prob. 37E Ch. 5.2 - Prob. 38E Ch. 5.2 - Prob. 39E Ch. 5.2 - Prob. 40E Ch. 5.2 - Prob. 41E Ch. 5.2 - Prob. 42E Ch. 5.2 - Prob. 43E Ch. 5.2 - 5-44. Determine the covariance and correlation for...Ch. 5.2 - Prob. 45E Ch. 5.2 - Prob. 46E Ch. 5.2 - Prob. 47E Ch. 5.3 - 5-48. Test results from an electronic circuit...Ch. 5.3 - Prob. 49E Ch. 5.3 - Prob. 50E Ch. 5.3 - 5-51. Four electronic ovens that were dropped...Ch. 5.3 - Prob. 52E Ch. 5.3 - Prob. 53E Ch. 5.3 - 5-54. In an acid-base titration, a base or acid is...Ch. 5.3 - Prob. 55E Ch. 5.3 - Prob. 56E Ch. 5.3 - Prob. 58E Ch. 5.3 - 5-59. If X and Y have a bivariate normal...Ch. 5.3 - 5-60. Show that the probability density function...Ch. 5.3 - Prob. 61E Ch. 5.4 - Prob. 62E Ch. 5.4 - 5-63. X and Y are independent, normal random...Ch. 5.4 - 5-64. Suppose that the random variable X...Ch. 5.4 - 5-65. A plastic casing for a magnetic disk is...Ch. 5.4 - 5-66. Making handcrafted pottery generally takes...Ch. 5.4 - 5-67. In the manufacture of electroluminescent...Ch. 5.4 - 5-68. The width of a casing for a door is normally...Ch. 5.4 - 5-69. An article in Knee Surgery Sports...Ch. 5.4 - 5-70. An automated filling machine fills...Ch. 5.4 - 5-71. The photoresist thickness in semiconductor...Ch. 5.4 - 5-72. Assume that the weights of individuals are...Ch. 5.4 - 5-73. Weights of parts are normally distributed...Ch. 5.4 - Prob. 74E Ch. 5.4 - Prob. 75E Ch. 5.4 - Prob. 76E Ch. 5.4 - Prob. 77E Ch. 5.4 - Prob. 78E Ch. 5.5 - 5-79. Suppose that X is a random variable with...Ch. 5.5 - 5-80. Let X be a binomial random variable with p =...Ch. 5.5 - 5-81. Suppose that X is a continuous random...Ch. 5.5 - 5-83. A random variable X has the probability...Ch. 5.5 - Prob. 84E Ch. 5.5 - Prob. 85E Ch. 5.5 - Prob. 86E Ch. 5.5 - Prob. 87E Ch. 5.5 - Prob. 88E Ch. 5.5 - 5-89. The computational time of a statistical...Ch. 5.5 - Prob. 90E Ch. 5.6 - Prob. 91E Ch. 5.6 - Prob. 92E Ch. 5.6 - 5-93. The geometric random variable X has...Ch. 5.6 - 5-94. The chi-squared random variable with k...Ch. 5.6 - Prob. 95E Ch. 5.6 - Prob. 96E Ch. 5.6 - Prob. 98E Ch. 5.6 - Prob. 99E Ch. 5.6 - 5-100. Suppose that Xi has a normal distribution...Ch. 5 - Prob. 101SE Ch. 5 - 5-102. The percentage of people given an...Ch. 5 - Prob. 103SE Ch. 5 - Prob. 104SE Ch. 5 - Prob. 105SE Ch. 5 - Prob. 106SE Ch. 5 - Prob. 107SE Ch. 5 - Prob. 108SE Ch. 5 - Prob. 109SE Ch. 5 - 5-110. The weight of adobe bricks for construction...Ch. 5 - Prob. 111SE Ch. 5 - Prob. 112SE Ch. 5 - Prob. 113SE Ch. 5 - Prob. 114SE Ch. 5 - Prob. 116SE Ch. 5 - Prob. 117SE Ch. 5 - Prob. 118SE Ch. 5 - Prob. 119SE Ch. 5 - 5-120. An order of 15 printers contains 4 with a...Ch. 5 - Prob. 121SE Ch. 5 - Prob. 122SE Ch. 5 - Prob. 123SE Ch. 5 - Prob. 124SE Ch. 5 - Prob. 125SE Ch. 5 - Prob. 126SE Ch. 5 - Prob. 127SE Ch. 5 - Prob. 128SE Ch. 5 - Prob. 129SE Ch. 5 - Prob. 130SE Ch. 5 - Prob. 131SE Ch. 5 - Prob. 132SE

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Find the probability that | X − Y | &lt; 0.5 using joint probability density function .

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Chapter 5 Solutions

Find the probability that | X − Y | < 0.5 using joint probability density function .