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 1 − x if 0 ≤ x < 1 0 if 1 ≤ x ≤ 2 x − 2 if 2 ≤ x ≤ 3 0 if x > 3 . Obtain the cumulative distribution function for each interval as follows. For ( − ∞ , 0 ) , F ( x ) = ∫ − ∞ x 0 d t = 0 . In case of [ 0 , 1 ] , the value of F ( x ) is, F ( x ) = ∫ − ∞ x f ( t ) d t = ∫ − ∞ 0 0 d t + ∫ 0 x ( 1 − t ) d t = ( t − t 2 2 ) 0 x = x − 1 2 x 2 In case of [ 1 , 2 ] , the value of F ( x ) is, F ( x ) = ∫ − ∞ x f ( t ) d t = ∫ − ∞ 0 0 d t + ∫ 0 1 ( 1 − t ) d t + ∫ 1 x 0 d t = ( t − 1 2 t 2 ) 0 1 = 1 2 In case of ( 2 , 3 ] , the value of F ( x ) is, F ( x ) = ∫ − ∞ x f ( t ) d t = ∫ − ∞ 0 0 d t + ∫ 0 1 ( 1 − t ) d t + ∫ 1 2 0 d t + ∫ 2 x ( t − 2 ) d t = ∫ 0 1 ( 1 − t ) d t + ∫ 2 x ( t − 2 ) d t = ( t − t 2 2 ) 0 1 +

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 33E

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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 32E

Chapter 11.3, Problem 34E

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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 by using graph and check the uniqueness.

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