Explanation Calculation: It is given that for completing one task, the maximum time is 2.5 days. The proportion completion time follows a beta distribution with the parameters α = 2 , β = 3 . Beta distribution: If X follows beta distribution then the probability density function of X is, f ( x ) = Γ ( α + β ) Γ ( α ) Γ ( β ) x α − 1 ( 1 − x ) β − 1 , for x in [ 0 , 1 ] , α , β > 0 , with mean E ( X ) = α α + β and variance V ( X ) = α β ( α + β ) 2 ( α + β + 1 ) . Let X be the completion proportion of the maximum time. Here, the proportion is 2 days 2.5 days = 0.8 The probability that the given task requires more than two days to complete: P ( X > 0.8 ) = ∫ 0.8 1 Γ ( α + β ) Γ ( α ) Γ ( β ) x α − 1 ( 1 − x ) β − 1 d x = Γ ( 2 + 3 ) Γ ( 2 ) Γ ( 3 ) ∫ 0.8 1 x 2 − 1 ( 1 − x ) 3 − 1 d x , substituting , α = 2 , β = 3 = Γ ( 5 ) Γ ( 2 ) Γ ( 3 ) ∫ 0

Applied Statistics and Probability for Engineers

6th Edition

ISBN: 9781118802250

Author: Montgomery

Publisher: WILEY

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Chapter 4.12, Problem 189E

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Find the probability that the given task requires more than two days to complete.

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Chapter 4.12, Problem 188E

Chapter 4.12, Problem 190E

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Applied Statistics and Probability for Engineers

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