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

Chapter 5.2, Problem 40E

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Applied Statistics and Probability for Engineers

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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.