Explanation Definition used: “A median is a value of the random variable that divides the area under the graph of the probability density function into two equal parts. That is, if m is a median, then m must satisfy the equation P ( X ≤ m ) = 1 2 .” Calculation: From the given figure, the probability density function is f ( x ) = { 0 if x < 0 0.5 if 0 ≤ x ≤ 1 0 if 1 < x < 2 0.5 if 2 ≤ x ≤ 3 0 3 < x . Use the definition of the cumulative distribution function to obtain F ( x ) as follows. For x < 0 , F ( x ) = P ( X < 0 ) = ∫ − ∞ x 0 d t [ ∵ f ( x ) = 0 for x < 0 ] = 0 Hence, the cumulative distribution function for x < 0 is F ( x ) = 0 . For 0 ≤ x ≤ 1 , F ( x ) = P ( X ≤ x ) = ∫ − ∞ x f ( t ) d t = ∫ − ∞ 0 0 d t + ∫ 0 x 0.5 d t On further simplification, F ( x ) = ∫ 0 x 0.5 d t = 0.5 ∫ 0 x d t = 0.5 [ t ] 0 x = 0.5 x Hence, the cumulative distribution function for 0 ≤ x ≤ 1 is F ( x ) = 0.5 x . For 1 ≤ x ≤ 2 , F ( x ) = P ( X ≤ x ) = ∫ − ∞ x f ( t ) d t = ∫ − ∞ 0 0 d t + ∫ 0 1 0.5 d t + ∫ 1 x 0 d t On further simplification, F ( x ) = ∫ 0 1 0.5 d t = 0.5 ∫ 0 x d t = 0.5 [ t ] 0 1 = 0.5 Hence, the cumulative distribution function for 1 ≤ x ≤ 2 is F ( x ) = 0.5 . For 2 ≤ x ≤ 3 , F ( x ) = P ( X ≤ x ) = ∫ − ∞ x f ( t ) d t = ∫ − ∞ 0 0 d t + ∫ 0 1 0

Calculus for Business, Economics, Life Sciences, and Social Sciences - Boston U.

15th Edition

ISBN: 9781323047620

Author: Barnett, Ziegler, Byleen

Publisher: Pearson Education

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Chapter 11.3, Problem 2ED

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Whether the median of the probability density function f exists and if so check the uniqueness.

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Chapter 11.3, Problem 1ED

Chapter 11.3, Problem 1E

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Chapter 11 Solutions

Calculus for Business, Economics, Life Sciences, and Social Sciences - Boston U.

Ch. 11.1 - Evaluate the following, if it converges: 3dx(x1)2.Ch. 11.1 - Prob. 2MP Ch. 11.1 - Prob. 3MP Ch. 11.1 - Prob. 4MP Ch. 11.1 - Prob. 5MP Ch. 11.1 - Prob. 6MP Ch. 11.1 - Prob. 1ED Ch. 11.1 - Prob. 2ED Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E

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Whether the median of the probability density function f exists and if so check the uniqueness.

Solution Summary: The author analyzes whether the median of the probability density function f exists and if so check the uniqueness.

Whether the median of the probability density function f exists and if so check the uniqueness.

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Chapter 11 Solutions