Explanation Calculation: Consider that Y 1 and Y 2 are two random variables with means μ 1 and μ 2 , respectively. The covariance of Y 1 and Y 2 is defined as, Cov ( Y 1 , Y 2 ) = E [ ( Y 1 − μ 1 ) ( Y 2 − μ 2 ) ] = E ( Y 1 Y 2 ) − E ( Y 1 ) E ( Y 2 ) . The random variables Y 1 and Y 2 are said to be independent if ad only if Cov ( Y 1 , Y 2 ) = 0 , that is, E ( Y 1 Y 2 ) = E ( Y 1 ) E ( Y 2 ) . The two continuous random variables Y 1 and Y 2 are said to be independent if and only if f ( y 1 , y 2 ) = f 1 ( y 1 ) f 2 ( y 2 ) , where f ( y 1 , y 2 ) is the joint probability density function of Y 1 and Y 2 and f 1 ( y 1 ) , f 2 ( y 2 ) are the marginal probability density functions of Y 1 and Y 2 , respectively. Consider that Y 1 and Y 2 are two continuous real valued random variables with joint probability density function of f ( y 1 , y 2 ) . Then, the marginal probability functions of Y 1 and Y 2 are defined as, f 1 ( y 1 ) = ∫ − ∞ ∞ f ( y 1 , y 2 ) d y 2 and f 2 ( y 2 ) = ∫ − ∞ ∞ f ( y 1 , y 2 ) d y 1 . The range of Y 1 and Y 2 are given as 0 ≤ y 1 ≤ y 2 ≤ 1 . Hence, the range of Y 1 is 0 ≤ y 1 ≤ y 2 and the range of Y 2 is y 1 ≤ y 2 ≤ 1 . Hence, the marginal probability density function for Y 1 is calculated below: f 1 ( y 1 ) = ∫ y 1 1 6 ( 1 − y 2 ) d y 2 = 6 [ ( 1 − y 2 ) 2 2 ] 1 y 1 = 3 [ ( 1 − y 1 ) 2 − ( 1 − 1 ) 2 ] = 3 ( 1 − y 1 ) 2 Thus, the marginal density function for Y 1 is f 1 ( y 1 ) = 3 ( 1 − y 1 ) 2 , for 0 ≤ y 1 ≤ 1 . In similar way, the marginal probability density function for Y 2 is calculated below: f 2 ( y 2 ) = ∫ 0 y 2 6 ( 1 − y 2 ) d y 1 = 6 ( 1 − y 2 ) [ y 1 ] 0 y 2 = 6 ( 1 − y 2 ) [ y 2 − 0 ] = 6 y 2 ( 1 − y 2 ) . Thus, the marginal density function for Y 2 is f 2 ( y 2 ) = 6 y 2 ( 1 − y 2 ) , 0 ≤ y 2 ≤ 1 . The expectation of Y 1 is calculated as follows: E ( Y 1 ) = ∫ 0 1 y 1 f 1 ( y 1 ) d y 1 = ∫ 0 1 ( y 1 ) [ 3 ( 1 − y 1 ) 2 ] d y 1 = 3 { ∫ 0 1 y 1 d y 1 − 2 ∫ 0 1 y 1 2 d y 1 + ∫ 0 1 y 1 3 d y 1 } = 3 { [ y 1 2 2 ] 0 1 − 2 [ y 1 3 3 ] 0 1 + [ y 1 4 4 ] 0 1 } = 3 { [ 1 2 − 0 ] − 2 [ 1 3 − 0 ] + [ 1 4 − 0 ] } = 3 × 1 12 = 1 4

Mathematical Statistics with Applications

7th Edition

ISBN: 9781133384380

Author: Dennis Wackerly; William Mendenhall; Richard L. Scheaffer

Publisher: Cengage Learning US

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Question

Chapter 5.7, Problem 92E

To determine

Find the value of Cov(Y1,Y2).

Explain whether Y1 and Y2 are independent.

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Chapter 5.7, Problem 91E

Chapter 5.7, Problem 93E

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