Explanation Given: The number of new subscribers (in thousands) to the service next year in Foxboro is X and the number of new subscribers (in thousands) next year in Sharon is Y . The joint probability density function defined on the interval D = { ( x , y ) | 0 ≤ x ≤ 5 ; 0 ≤ y ≤ 4 } is as follows: f ( x , y ) = 9 4000 x y 25 − x 2 ( 4 − y ) Concept used: The joint probability density function of the random variables X and Y in an interval D can be expresses as below: ∬ D f ( x , y ) d A = 1 The probability of random variable X and Y that lie in a region R can be calculated as below: P [ ( x , y ) in R ] = ∬ R f ( x , y ) d A Calculation: Consider P as the probability that the number of new subscribers in Foxboro will be between 2000 and 2500, and the number of new subscribers in Sharon will be between 1000 and 2000. The random variable is in thousands. Therefore, the limit for x is 2 to 2.5 and limit for y is 1 and 2. The probability P can be calculated as follows: P ( 2 ≤ X ≤ 2.5 ; 1 ≤ Y ≤ 2 ) = ∫ 1 2 ∫ 2 2.5 9 4000 x y 25 − x 2 ( 4 − y ) d x d y = 9 4000 ∫ 1 2 ( ∫ 2 2.5 ( x 25 − x 2 d x ) ) y ( 4 − y ) d y Label the above equation as equation ( 1 ) . P ( 2 ≤ X ≤ 2.5 ; 1 ≤ Y ≤ 2 ) = 9 4000 ∫ 1 2 ( ∫ 2 2.5 ( x 25 − x 2 d x ) ) y ( 4 − y ) d y ...... ( 1 ) Consider 25 − x 2 as t in the above equation. 25 − x 2 = t ...... ( 2 ) Differentiate the above equation with respect to x as shown below: d d x ( 25 − x 2 ) = d d x t − 2 x = d t d x − 2 x d x = d t x d x = − 1 2 d t If the limit in x is 2 to 2

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

10th Edition

ISBN: 9781305657861

Author: Soo T. Tan

Publisher: Cengage Learning

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Question

Chapter 10.1, Problem 57E

To determine

The probability that the number of new subscribers next year in Foxboro will be between 2000 and 2500, and the number of new subscribers in Sharon will be between 1000 and 2000.

Expert Solution & Answer

Chapter 10 Solutions

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

Ch. 10.1 - Define the following terms in your own words: a....Ch. 10.1 - Prob. 2CQ Ch. 10.1 - Prob. 3CQ Ch. 10.1 - Prob. 1E Ch. 10.1 - Prob. 2E Ch. 10.1 - Prob. 3E Ch. 10.1 - Prob. 4E Ch. 10.1 - Prob. 5E Ch. 10.1 - Prob. 6E Ch. 10.1 - Prob. 7E

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The probability that the number of new subscribers next year in Foxboro will be between 2000 and 2500, and the number of new subscribers in Sharon will be between 1000 and 2000.

Solution Summary: The author explains that the number of new subscribers in Foxboro and Sharon will be between 2000 and 2000. The joint probability density function of the random variables in a region R can be calculated.

The probability that the number of new subscribers next year in Foxboro will be between 2000 and 2500, and the number of new subscribers in Sharon will be between 1000 and 2000.

Chapter 10 Solutions