Explanation Given: The probability density function is f ( t ) = 9 4 t 3 and t varies from 1 to 3 ( 1 ≤ t ≤ 3 ) . Concept used: For a probability density function f defined on the interval [ a , b ] associated with a continuous random variable X , the expected value μ of X is written as follows. μ = ∫ a b x f ( x ) d x ...... (1) The variance of X is written as follows. Var ( X ) = ∫ a b ( x − μ ) 2 f ( x ) d x ...... (2) The standard deviation of X is written as follows. σ = Var ( X ) ...... (3) Calculation: The probability density function is given as f ( t ) = 9 4 t 3 over the interval [ 1 , 3 ] . Substitute 1 for a , 3 for b and 9 4 t 3 for f ( t ) in equation ( 1 ) to obtain the expected reaction time as shown below: μ = ∫ 1 3 t ( 9 4 t 3 ) d t = 9 4 ∫ 1 3 ( 1 t 2 ) d t = 9 4 ⋅ [ − 1 t ] 1 3 The above expression can be further simplified as follows. μ = 9 4 ⋅ [ − 1 3 − ( − 1 1 ) ] = 9 4 ⋅ [ − 1 3 + 1 ] = 9 4 ⋅ [ 0

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

10th Edition

ISBN: 9781305657861

Author: Soo T. Tan

Publisher: Cengage Learning

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Chapter 10.2, Problem 17E

To determine

The expected reaction time for a motorist and also find the variance and standard deviation of the probability density function f(t)=94t3 over the interval [1,3].

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Chapter 10.2, Problem 16E

Chapter 10.2, Problem 18E

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Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

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The expected reaction time for a motorist and also find the variance and standard deviation of the probability density function f ( t ) = 9 4 t 3 over the interval [ 1 , 3 ] .

Solution Summary: The author explains the probability density function f(t)=94t

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The expected reaction time for a motorist and also find the variance and standard deviation of the probability density function f(t)=94t3 over the interval [1,3].

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Chapter 10 Solutions