To determine Find the mean and variance of the random variable in Exercise 4-14. Explanation Calculation: The given information is that the triangular probability density function is, f ( x ) = { 0.025 x − 0.0375 for 30 < x < 50 − 0.025 x + 0.0875 for 50 < x < 70 The total probability of the given function is, Probability = ∫ 30 50 ( 0.025 x − 0.0375 ) d x + ∫ 50 70 ( − 0.025 x + 0.0875 ) d x = 0.025 ( x 2 2 ) 30 50 − 0.0375 ( x ) 30 50 − 0.025 ( x 2 2 ) 50 70 + 0.0875 ( x ) 50 70 = 0.025 ( 50 2 − 30 2 − ( 70 2 − 50 2 ) 2 ) − 0.0375 ( 50 − 30 ) + 0.0875 ( 70 − 50 ) = 0.025 ( 5 , 000 − 5 , 800 2 ) − 0.0375 ( 20 ) + 0.0875 ( 20 ) = − 10 − 0.75 + 1.75 = − 9 Here, it is observed that, the probability of the given density function is not equal to 1. Thus, the given probability density function is wrong. The general form of the triangular probability density function is, f ( x ) = { 2 ( x − a ) ( b − a ) ( m − a ) a < x < m 2 ( b − x ) ( b − a ) ( b − m ) m < x < b From the given information, the limits a , m and b are 30, 50 and 70. Now, substitute the limits a , m and b in general form of the triangular probability density function. f ( x ) = { 2 ( x − 30 ) ( 70 − 30 ) ( 50 − 30 ) 30 < x < 50 2 ( 70 − x ) ( 70 − 30 ) ( 70 − 50 ) 50 < x < 70 = { 2 ( x − 30 ) ( 40 ) ( 20 ) 30 < x < 50 2 ( 70 − x ) ( 40 ) ( 20 ) 50 < x < 70 = { x − 30 400 30 < x < 50 70 − x 400 50 < x < 70 = { 0.0025 x − 0.075 30 < x < 50 − 0.0025 x + 0.175 50 < x < 70 The total probability of the triangular probability density function based on given interval is, Probability = ∫ 30 50 ( 0.0025 x − 0.075 ) d x + ∫ 50 70 ( − 0.0025 x + 0.175 ) d x = 0.0025 ( x 2 2 ) 30 50 − 0.075 ( x ) 30 50 − 0.0025 ( x 2 2 ) 50 70 + 0.175 ( x ) 50 70 = 0.0025 ( 50 2 − 30 2 − ( 70 2 − 50 2 ) 2 ) − 0

Applied Statistics and Probability for Engineers

6th Edition

ISBN: 9781118539712

Author: Douglas C. Montgomery

Publisher: WILEY

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Chapter 4.4, Problem 42E

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Find the mean and variance of the random variable in Exercise 4-14.

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Chapter 4.4, Problem 41E

Chapter 4.4, Problem 43E

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Applied Statistics and Probability for Engineers

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