a. Explanation Let X be the number of oranges. In this context, x oranges were selected out of 3 oranges in ( 3 x ) ways. Let Y be the number of apples and y apples were selected out of 2 apples in ( 2 y ) ways. It is known that x oranges and y apples have already been selected in a sample of 4 fruits. Therefore, there are 4 − x − y bananas in a sack. Thus, the possible ways to select 4 − x − y bananas out of 3 bananas is ( 3 4 − x − y ) ways. With reference to Exercise 3.39, the joint distribution for X and Y is given below: f ( x , y ) = ( 3 x ) ( 2 y ) ( 3 4 − x − y ) ( 8 4 ) , x = 0 , 1 , 2 , 3 ; y = 0 , 1 , 2 The table represents the value of joint distribution for X and Y : For x =0, y =0, f ( 0 , 0 ) = ( 3 0 ) ( 2 0 ) ( 3 4 − 0 − 0 ) ( 8 4 ) = 0 In similar way, the remaining values are obtained. f(x, y) x y 0 1 2 3 Total 0 0 3 70 9 70 3 70 15 70 1 2 70 18 70 18 70 2 70 40 70 2 3 70 9 70 3 70 0 15 70 Total 5 70 30 70 30 70 5 70 1 Expected value or mean of X : μ g ( X , Y ) = E ( g ( X , Y ) ) = { ∑ x ∑ y g ( x , y ) f ( x , y ) , For discrete ∫ − ∞ ∞ ∫ − ∞ ∞ g ( x , y ) f ( x , y ) d x d y , For continuous In this context, the random variable X is discrete b. To determine Obtain the value of μ x − μ y .

Probability and Statistics for Engineers and Scientists

9th Edition

ISBN: 9780321629111

Author: Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, Keying Ye

Publisher: Prentice Hall

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Question

Chapter 4.1, Problem 24E

To determine

Obtain the value of E(X2Y−2XY).

To determine

Obtain the value of μx−μy.

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Chapter 4.1, Problem 23E

Chapter 4.1, Problem 25E

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Client 1 Weight before diet (pounds) Weight after diet (pounds) 128 120 2 131 123 3 140 141 4 178 170 5 121 118 6 136 136 7 118 121 8 136 127