Explanation Definition used: 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: The probability density function is f ( x ) = { 3 x − 3 x 5 if 0 ≤ x ≤ 1 0 Otherwise . Initially consider the case when x < 0 . Use the definition of the cumulative distribution function to obtain the value of F ( x ) . 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 . In the same manner obtain the cumulative distribution function for x ≥ 0 as, F ( x ) = P ( X ≤ x ) = ∫ − ∞ x f ( t ) d t = ∫ − ∞ 0 0 d t + ∫ 0 x ( 3 t − 3 t 5 ) d t On further simplification, F ( x ) = ∫ 0 x ( 3 t − 3 t 5 ) d t = ( 3 t 2 2 − t 6 2 ) 0 x = 3 x 2 2 − x 6 2 So, the cumulative distribution function for x ≥ 0 is F ( x ) = 3 x 2 2 − x 6 2

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 42E

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To find: The median of the probability density function f(x)={3x−3x5 if 0≤x≤10 Otherwise .

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Chapter 11.3, Problem 41E

Chapter 11.3, Problem 43E

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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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The median of the probability density function f ( x ) = { 3 x − 3 x 5 if 0 ≤ x ≤ 1 0 Otherwise .

Solution Summary: The author explains the probability density function's median value, which is 0.59.

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To find: The median of the probability density function f(x)={3x−3x5 if 0≤x≤10 Otherwise .

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Chapter 11 Solutions