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 is expressed as shown below: ∬ D f ( x , y ) d A = 1 The probability of random variable X and Y that lie in a region R is calculated as follows: 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 less than 3000 and the number of new subscribers in Sharon will be less than 2000. The random variable is in thousands. So, the new subscribers in Foxboro will be less than 3 and the number of new subscribers in Sharon will be less than 2. The probability P can be calculated as follows: P ( 0 ≤ X ≤ 3 ; 0 ≤ Y ≤ 2 ) = ∫ 0 2 ∫ 0 3 9 4000 x y 25 − x 2 ( 4 − y ) d x d y = 9 4000 ∫ 0 2 ( ∫ 0 3 ( 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 ∫ 0 2 ( ∫ 0 3 ( 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 follows: 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 0 to 3 then the limit in t can be calculated as below

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Chapter 10.1, Problem 58E

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The probability that the number of new subscribers next year in Foxboro will be less than 3000 and the number of new subscribers in Sharon will be less than 2000.

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Chapter 10.1, Problem 57E

Chapter 10.1, Problem 59E

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The probability that the number of new subscribers next year in Foxboro will be less than 3000 and the number of new subscribers in Sharon will be less than 2000.

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The probability that the number of new subscribers next year in Foxboro will be less than 3000 and the number of new subscribers in Sharon will be less than 2000.

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