Explanation Definition used: The cumulative distribution function is F ( x ) = ∫ − ∞ x f ( t ) d t , where f ( t ) represents the probability density function. Median: It is the value of the random variable and which divides the area under the graph of the probability density function into 2 equal parts. If m satisfy the condition P ( X ≤ m ) = 1 2 , then m is the median. Calculation: From the given graph, the probability density function is, f ( x ) = { 0 if x < 0 2 − 4 x if 0 ≤ x < 1 2 0 if 1 2 ≤ x ≤ 5 2 4 x -10 if 5 2 < x ≤ 3 0 if x > 3 . For a cumulative distribution function, F ( x ) = ∫ − ∞ x f ( t ) d t . For ( − ∞ , 0 ) , F ( x ) = ∫ − ∞ x 0 d t = 0 . In case of [ 0 , 1 2 ] , compute the value of F ( x ) . F ( x ) = ∫ − ∞ x f ( t ) d t = ∫ − ∞ 0 0 d t + ∫ 0 x ( 2 − 4 t ) d t = ( 2 t − 2 t 2 ) 0 x = 2 x − 2 x 2 In case of [ 1 2 , 5 2 ] , compute the value of F ( x ) . F ( x ) = ∫ − ∞ x f ( t ) d t = ∫ − ∞ 0 0 d t + ∫ 0 1 2 ( 2 − 4 t ) d t + ∫ 1 2 x 0 d t = ( 2 t − 2 t 2 ) 0 1 2 = 1 2 In case of ( 5 2 , 3 ] , compute the value of F ( x ) . F ( x ) = ∫ − ∞ x f ( t ) d t = ∫ − ∞ 0 0 d t + ∫ 0 1 2 ( 2 − 4 t ) d t + ∫ 1 2 5 2 0 d t + ∫ 5 2 x ( 4 t − 10 ) d t = ∫ 0 1 2 ( 2 − 4 t ) d t + ∫ 5 2 x ( 4 t − 10 ) d t = ( 2 t − 2 t 2 ) 0 1 2 + ( 2 t 2 − 10 t ) 5 2 x On further simplification, F ( x ) = ( 1 − 1 2 ) + ( 2 x 2 − 10 x − 25 2 + 25 ) = 1 2 + ( 2 x 2 − 10 x + 25 2 ) = 2 x 2 − 10 x + 26 2 = 2 x 2 − 10 x + 13 For x > 3 , compute the value of F ( x )

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

14th Edition

ISBN: 9780134668574

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Christopher J. Stocker

Publisher: PEARSON

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Chapter 11.3, Problem 34E

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To find: The median of the probability density function by using graph and check the uniqueness.

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Chapter 11.3, Problem 33E

Chapter 11.3, Problem 35E

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Chapter 11 Solutions

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

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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The median of the probability density function by using graph and check the uniqueness.

Solution Summary: The author calculates the median of the probability density function by using graph and checking the uniqueness.

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Chapter 11 Solutions