Explanation Definition used: If f is a probability density function, then the associated cumulative distribution function F is F ( x ) = P ( X ≤ x ) = ∫ − ∞ ∞ f ( t ) d t . Formula used: Let f ( x ) be the probability density function for a continuous random variable X . Median (m): “If m is the median, then m must satisfy P ( X ≤ m ) = 1 2 ”. Calculation: The given function is f ( x ) = { 1 x if 1 ≤ x ≤ e 0 othewise . Use the definition of the cumulative distribution function to obtain F ( x ) as follows. For x < 1 , F ( x ) = P ( X < 1 ) = ∫ − ∞ x 0 d t [ ∵ f ( x ) = 0 for x < 1 ] = 0 Hence, the cumulative distribution function for x < 1 is F ( x ) = 0 . For 1 ≤ x < e , F ( x ) = P ( X ≤ x ) = ∫ − ∞ x f ( t ) d t = ∫ − ∞ 1 0 d t + ∫ 1 x ( 1 t ) d t On further simplification, F ( x ) = ∫ 1 x ( 1 t ) d t = ( ln t ) 1 x = ( ln x − ln 1 ) = ln x So, the cumulative distribution function for 1 ≤ x < e is F ( x ) = ln 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 23E

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To find: The median for the given function.

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Chapter 11.3, Problem 22E

Chapter 11.3, Problem 24E

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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 for the given function.

Solution Summary: The author explains how to find the median for the given function. The median of the function is 1.649.

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Chapter 11 Solutions