Explanation Given info: The joint probability density function for X and Y is given as f X Y ( x , y ) = c x y over the range 0 < x < 3 and 0 < y < x . Calculation: The joint probability density function of the continuous random variable X, 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 ) = 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 < 3 and 0 < y < x , respectively. Thus, ∫ − ∞ ∞ ∫ − ∞ ∞ f XY ( x , y ) = 1 ∫ 0 3 ∫ 0 x c x y d y d x = 1 c ∫ 0 3 [ x ∫ 0 x y d y ] d x = 1 c ∫ 0 3 [ x [ y 2 2 ] 0 x ] d x = 1 c 2 ∫ 0 3 x [ x 2 − 0 ] d x = 1 c 2 ∫ 0 3 x 3 d x = 1 c 8 [ x 4 ] 0 3 = 1 c 8 ( 4 3 − 0 ) = 8 c = 8 81 Hence, c = 8 81 . Now, the joint probability density function is f ( x , y ) = 8 81 x y for 0 < x < 3 and 0 < y < x . The expected value of a function of two random variables is defined as, E ( h ( X , Y ) ) = { ∑ x ∑ y h ( x , y ) f X Y ( x , y ) , X , Y dicrete ∫ x ∫ y h ( x , y ) f X Y ( x , y ) d y d x , X , Y continuous Variance of a continuous random variable X is defined as, V ( X ) = E ( X 2 ) − E 2 ( X ) = ∫ x ∫ y x 2 f XY ( x , y ) d y d x − ( ∫ x ∫ y x f XY ( x , y ) d y d x ) 2 , Thus, the mean of the random variable X is, E ( X ) = ∫ x ∫ y x f XY ( x , y ) d y d x = ∫ 0 3 ∫ 0 x x 8 81 x y d y d x = 8 81 ∫ 0 3 [ x 2 ∫ 0 x y d y ] d x = 8 81 ∫ 0 3 [ x 2 [ y 2 2 ] 0 x ] d x = 8 81 1 2 ∫ 0 3 x 2 x 2 d x = 4 81 ∫ 0 3 x 4 d x = 4 81 [ x 5 5 ] 0 3 = 4 81 ( 3 5 5 − 0 ) = 4 81 243 5 = 2.4 Now, E ( X 2 ) = ∫ x ∫ y x 2 f XY ( x , y ) d y d x = ∫ 0 3 ∫ 0 x 8 81 x 3 y d y d x = 8 81 ∫ 0 3 [ x 3 ∫ 0 x y d y ] d x = 8 81 ∫ 0 3 [ x 3 [ y 2 2 ] 0 x ] d x = 8 81 1 2 ∫ 0 3 x 3 x 2 d x = 4 81 ∫ 0 3 x 5 d x = 4 81 [ x 6 6 ] 0 3 = 4 81 ( 3 6 6 − 0 ) = 4 81 729 6 = 6 Hence, the variance of X is, V ( X ) = E ( X 2 ) − E 2 ( X ) = 6 − ( 2

Applied Statistics and Probability for Engineers

6th Edition

ISBN: 9781118539712

Author: Douglas C. Montgomery

Publisher: WILEY

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Chapter 5.2, Problem 39E

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Find the value of c for the following joint probability density function.

Find the covariance and correlation for the following joint probability density function.

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Chapter 5.2, Problem 38E

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Applied Statistics and Probability for Engineers

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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 value of c for the following joint probability density function.

Find the covariance and correlation for the following joint probability density function.