Explanation Given: The function f ( x ) = x ( x 2 + 1 ) 3 2 and specified interval 0 ≤ x ≤ ∞ . Definition used: “A probability density function of a random variable X in an interval I , where I may be bounded or unbounded is a nonnegative function f having the following properties. 1. The total area of the region under the graph of f is equal to 1 2. The probability that an observed value of the random variable X lies in the interval [ a , b ] is P ( a ≤ X ≤ b ) = ∫ a b f ( x ) d x .” Proof: The given function f ( x ) = x ( x 2 + 1 ) 3 2 is non negative on [ 0 , ∞ ] . The total area of the region under the graph of f is computed follows, Integrate the function f ( x ) = x ( x 2 + 1 ) 3 2 on [ 0 , ∞ ] and obtain the area, ∫ 0 ∞ x ( x 2 + 1 ) 3 2 d x = ∫ 0 ∞ x ( x 2 + 1 ) 3 2 d x Let x 2 + 1 = t , then 2 x d x = d t x d x = d t 2 If x = 0 , then t = 1

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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Chapter 10.1, Problem 9E

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Chapter 10.1, Problem 8E

Chapter 10.1, Problem 10E

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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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Chapter 10 Solutions